{"id":3229,"date":"2018-07-31T13:44:06","date_gmt":"2018-07-31T13:44:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/?post_type=chapter&#038;p=3229"},"modified":"2018-07-31T13:47:33","modified_gmt":"2018-07-31T13:47:33","slug":"the-ellipse","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/the-ellipse\/","title":{"raw":"The Ellipse","rendered":"The Ellipse"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Write equations of ellipses in standard form.<\/li>\r\n \t<li>Graph ellipses centered at the origin.<\/li>\r\n \t<li>Graph ellipses not centered at the origin.<\/li>\r\n \t<li>Solve applied problems involving ellipses.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Figure_10_01_001\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151211\/CNX_Precalc_Figure_10_01_001n.jpg\" alt=\"\" width=\"488\" height=\"324\" \/> <strong>Figure 1. <\/strong>The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id693728\">Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_001\">(Figure)<\/a>, is such a room.[footnote]Architect of the Capitol. http:\/\/www.aoc.gov. Accessed April 15, 2014.[\/footnote] It is an oval-shaped room called a <em>whispering chamber<\/em> because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.<\/p>\r\n\r\n<div id=\"fs-id1297573\" class=\"bc-section section\">\r\n<h3>Writing Equations of Ellipses in Standard Form<\/h3>\r\n<p id=\"fs-id1332910\">A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_002\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_10_01_002\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"976\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151214\/CNX_Precalc_Figure_10_01_002.jpg\" alt=\"\" width=\"976\" height=\"441\" \/> <strong>Figure 2.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1332535\">Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).<\/p>\r\n<p id=\"fs-id1133644\">We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_003\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_10_01_003\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151217\/CNX_Precalc_Figure_10_01_003.jpg\" alt=\"\" width=\"487\" height=\"560\" \/> <strong>Figure 3.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1364494\">Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_004\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_10_01_004\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151227\/CNX_Precalc_Figure_10_01_004.jpg\" alt=\"\" width=\"731\" height=\"366\" \/> <strong>Figure 4.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1337244\">In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the <em>x<\/em>- and <em>y<\/em>-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.<\/p>\r\n<p id=\"fs-id1194638\">To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.<\/p>\r\n\r\n<div id=\"fs-id1366768\" class=\"bc-section section\">\r\n<h4>Deriving the Equation of an Ellipse Centered at the Origin<\/h4>\r\n<p id=\"fs-id1297787\">To derive the equation of an <span class=\"no-emphasis\">ellipse<\/span> centered at the origin, we begin with the foci[latex]\\,\\left(-c,0\\right)\\,[\/latex]and[latex]\\,\\left(c,0\\right).\\,[\/latex]The ellipse is the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]such that the sum of the distances from[latex]\\,\\left(x,y\\right)\\,[\/latex]to the foci is constant, as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_014\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_10_01_014\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151234\/CNX_Precalc_Figure_10_01_014.jpg\" alt=\"\" width=\"487\" height=\"274\" \/> <strong>Figure 5.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1226529\">If[latex]\\,\\left(a,0\\right)\\,[\/latex]is a <span class=\"no-emphasis\">vertex<\/span> of the ellipse, the distance from[latex]\\,\\left(-c,0\\right)\\,[\/latex]to[latex]\\,\\left(a,0\\right)\\,[\/latex]is[latex]\\,a-\\left(-c\\right)=a+c.\\,[\/latex]The distance from[latex]\\,\\left(c,0\\right)\\,[\/latex]to[latex]\\,\\left(a,0\\right)\\,[\/latex]is[latex]\\,a-c[\/latex]. The sum of the distances from the <span class=\"no-emphasis\">foci<\/span> to the vertex is<\/p>\r\n\r\n<div id=\"fs-id1162201\" class=\"unnumbered aligncenter\">[latex]\\left(a+c\\right)+\\left(a-c\\right)=2a[\/latex]<\/div>\r\n<p id=\"fs-id1347565\">If[latex]\\,\\left(x,y\\right)\\,[\/latex]is a point on the ellipse, then we can define the following variables:<\/p>\r\n\r\n<div id=\"fs-id1105711\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{d}_{1}=\\text{the distance from }\\left(-c,0\\right)\\text{ to }\\left(x,y\\right)\\hfill \\\\ {d}_{2}=\\text{the distance from }\\left(c,0\\right)\\text{ to }\\left(x,y\\right)\\hfill \\end{array}[\/latex]<\/div>\r\nBy the definition of an ellipse,[latex]\\,{d}_{1}+{d}_{2}\\,[\/latex]is constant for any point[latex]\\,\\left(x,y\\right)\\,[\/latex]on the ellipse. We know that the sum of these distances is[latex]\\,2a\\,[\/latex]for the vertex[latex]\\,\\left(a,0\\right).\\,[\/latex]It follows that[latex]\\,{d}_{1}+{d}_{2}=2a\\,[\/latex]for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.\r\n<div id=\"fs-id1106644\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }{d}_{1}+{d}_{2}=\\sqrt{{\\left(x-\\left(-c\\right)\\right)}^{2}+{\\left(y-0\\right)}^{2}}+\\sqrt{{\\left(x-c\\right)}^{2}+{\\left(y-0\\right)}^{2}}=2a\\hfill &amp; \\text{Distance formula}\\hfill \\\\ \\sqrt{{\\left(x+c\\right)}^{2}+{y}^{2}}+\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}=2a\\hfill &amp; \\text{Simplify expressions}\\text{.}\\hfill \\\\ \\text{ }\\sqrt{{\\left(x+c\\right)}^{2}+{y}^{2}}=2a-\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\hfill &amp; \\text{Move radical to opposite side}\\text{.}\\hfill \\\\ \\text{ }{\\left(x+c\\right)}^{2}+{y}^{2}={\\left[2a-\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\right]}^{2}\\hfill &amp; \\text{Square both sides}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}+2cx+{c}^{2}+{y}^{2}=4{a}^{2}-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}+{\\left(x-c\\right)}^{2}+{y}^{2}\\hfill &amp; \\text{Expand the squares}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}+2cx+{c}^{2}+{y}^{2}=4{a}^{2}-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}+{x}^{2}-2cx+{c}^{2}+{y}^{2}\\hfill &amp; \\text{Expand remaining squares}\\text{.}\\hfill \\\\ \\text{ }2cx=4{a}^{2}-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}-2cx\\hfill &amp; \\text{Combine like terms}\\text{.}\\hfill \\\\ \\text{ }4cx-4{a}^{2}=-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\hfill &amp; \\text{Isolate the radical}\\text{.}\\hfill \\\\ \\text{ }cx-{a}^{2}=-a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\hfill &amp; \\text{Divide by 4}\\text{.}\\hfill \\\\ \\text{ }{\\left[cx-{a}^{2}\\right]}^{2}={a}^{2}{\\left[\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\right]}^{2}\\hfill &amp; \\text{Square both sides}\\text{.}\\hfill \\\\ \\text{ }{c}^{2}{x}^{2}-2{a}^{2}cx+{a}^{4}={a}^{2}\\left({x}^{2}-2cx+{c}^{2}+{y}^{2}\\right)\\hfill &amp; \\text{Expand the squares}\\text{.}\\hfill \\\\ \\text{ }{c}^{2}{x}^{2}-2{a}^{2}cx+{a}^{4}={a}^{2}{x}^{2}-2{a}^{2}cx+{a}^{2}{c}^{2}+{a}^{2}{y}^{2}\\hfill &amp; \\text{Distribute }{a}^{2}.\\hfill \\\\ \\text{ }{a}^{2}{x}^{2}-{c}^{2}{x}^{2}+{a}^{2}{y}^{2}={a}^{4}-{a}^{2}{c}^{2}\\hfill &amp; \\text{Rewrite}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}\\left({a}^{2}-{c}^{2}\\right)+{a}^{2}{y}^{2}={a}^{2}\\left({a}^{2}-{c}^{2}\\right)\\hfill &amp; \\text{Factor common terms}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}{b}^{2}+{a}^{2}{y}^{2}={a}^{2}{b}^{2}\\hfill &amp; \\text{Set }{b}^{2}={a}^{2}-{c}^{2}.\\hfill \\\\ \\text{ }\\frac{{x}^{2}{b}^{2}}{{a}^{2}{b}^{2}}+\\frac{{a}^{2}{y}^{2}}{{a}^{2}{b}^{2}}=\\frac{{a}^{2}{b}^{2}}{{a}^{2}{b}^{2}}\\hfill &amp; \\text{Divide both sides by }{a}^{2}{b}^{2}.\\hfill \\\\ \\text{ }\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\hfill &amp; \\text{Simplify}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1373013\">Thus, the standard equation of an ellipse is[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1.[\/latex]This equation defines an ellipse centered at the origin. If[latex]\\,a&gt;b,[\/latex]the ellipse is stretched further in the horizontal direction, and if[latex]\\,b&gt;a,[\/latex] the ellipse is stretched further in the vertical direction.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1381696\" class=\"bc-section section\">\r\n<h4>Writing Equations of Ellipses Centered at the Origin in Standard Form<\/h4>\r\n<p id=\"fs-id1376358\">Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we\u2019ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.<\/p>\r\n<p id=\"fs-id1308221\">The key features of the <span class=\"no-emphasis\">ellipse<\/span> are its center, <span class=\"no-emphasis\">vertices<\/span>, <span class=\"no-emphasis\">co-vertices<\/span>, <span class=\"no-emphasis\">foci<\/span>, and lengths and positions of the <span class=\"no-emphasis\">major and minor axes<\/span>. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.<\/p>\r\n\r\n<div id=\"fs-id1343526\" class=\"textbox key-takeaways\">\r\n<h3>Standard Forms of the Equation of an Ellipse with Center (0,0)<\/h3>\r\n<p id=\"fs-id1389856\">The standard form of the equation of an ellipse with center[latex]\\,\\left(0,0\\right)\\,[\/latex]and major axis on the <em>x-axis<\/em> is<\/p>\r\n\r\n<div id=\"Equation_10_01_01\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/div>\r\n<p id=\"fs-id1321099\">where<\/p>\r\n\r\n<ul id=\"fs-id1321102\">\r\n \t<li>[latex]a&gt;b[\/latex]<\/li>\r\n \t<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(\u00b1a,0\\right)[\/latex]<\/li>\r\n \t<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(0,\u00b1b\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(\u00b1c,0\\right)[\/latex], where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_005\">(Figure)<\/a><strong>a<\/strong><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1327739\">The standard form of the equation of an ellipse with center[latex]\\,\\left(0,0\\right)\\,[\/latex]and major axis on the <em>y-axis<\/em> is<\/p>\r\n\r\n<div id=\"Equation_10_01_02\">[latex]\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1[\/latex]<\/div>\r\n<p id=\"fs-id1390177\">where<\/p>\r\n\r\n<ul id=\"fs-id1390180\">\r\n \t<li>[latex]a&gt;b[\/latex]<\/li>\r\n \t<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(0,\u00b1a\\right)[\/latex]<\/li>\r\n \t<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(\u00b1b,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(0,\u00b1c\\right)[\/latex], where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_005\">(Figure)<\/a><strong>b<\/strong><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1333061\">Note that the vertices, co-vertices, and foci are related by the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.<\/p>\r\n\r\n<div id=\"Figure_10_01_005\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"935\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151236\/CNX_Precalc_Figure_10_01_005-1.jpg\" alt=\"\" width=\"935\" height=\"440\" \/> <strong>Figure 6. <\/strong>(a) Horizontal ellipse with center[latex]\\,\\left(0,0\\right)\\,[\/latex](b) Vertical ellipse with center[latex]\\,\\left(0,0\\right)[\/latex][\/caption]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1369821\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1278660\"><strong>Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1278664\" type=\"1\">\r\n \t<li>Determine whether the major axis lies on the <em>x<\/em>- or <em>y<\/em>-axis.\r\n<ol id=\"fs-id1350073\" type=\"a\">\r\n \t<li>If the given coordinates of the vertices and foci have the form[latex]\\,\\left(\u00b1a,0\\right)\\,[\/latex]and[latex]\\,\\left(\u00b1c,0\\right)\\,[\/latex]respectively, then the major axis is the <em>x<\/em>-axis. Use the standard form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1.[\/latex]<\/li>\r\n \t<li>If the given coordinates of the vertices and foci have the form[latex]\\,\\left(0,\u00b1a\\right)\\,[\/latex]and[latex]\\,\\left(\u00b1c,0\\right),[\/latex]respectively, then the major axis is the <em>y<\/em>-axis. Use the standard form[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1.[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Use the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2},\\,[\/latex]along with the given coordinates of the vertices and foci, to solve for[latex]\\,{b}^{2}.[\/latex]<\/li>\r\n \t<li>Substitute the values for[latex]\\,{a}^{2}\\,[\/latex]and[latex]\\,{b}^{2}\\,[\/latex]into the standard form of the equation determined in Step 1.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_10_01_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1326575\">\r\n<div id=\"fs-id1326578\">\r\n<h3>Writing the Equation of an Ellipse Centered at the Origin in Standard Form<\/h3>\r\n<p id=\"fs-id1326583\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(\u00b18,0\\right)\\,[\/latex]and foci[latex]\\,\\left(\u00b15,0\\right)?\\,[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1370498\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1370498\"]\r\n<p id=\"fs-id1370498\">The foci are on the <em>x<\/em>-axis, so the major axis is the <em>x<\/em>-axis. Thus, the equation will have the form<\/p>\r\n\r\n<div id=\"fs-id1344890\" class=\"unnumbered aligncenter\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/div>\r\n<p id=\"fs-id1350382\">The vertices are[latex]\\,\\left(\u00b18,0\\right),[\/latex]so[latex]\\,a=8\\,[\/latex]and[latex]\\,{a}^{2}=64.[\/latex]<\/p>\r\n<p id=\"fs-id1339910\">The foci are[latex]\\,\\left(\u00b15,0\\right),[\/latex]so[latex]\\,c=5\\,[\/latex]and[latex]\\,{c}^{2}=25.[\/latex]<\/p>\r\n<p id=\"fs-id1355048\">We know that the vertices and foci are related by the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,{b}^{2},[\/latex] we have:<\/p>\r\n\r\n<div id=\"fs-id1362221\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}{c}^{2}={a}^{2}-{b}^{2}\\hfill &amp; \\hfill \\\\ 25=64-{b}^{2}\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Substitute for }{c}^{2}\\text{ and }{a}^{2}.\\hfill \\\\ {b}^{2}=39\\hfill &amp; \\text{Solve for }{b}^{2}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1369043\">Now we need only substitute[latex]\\,{a}^{2}=64\\,[\/latex]and[latex]\\,{b}^{2}=39\\,[\/latex]into the standard form of the equation. The equation of the ellipse is[latex]\\,\\frac{{x}^{2}}{64}+\\frac{{y}^{2}}{39}=1.[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1304230\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_10_01_01\">\r\n<div id=\"fs-id1340710\">\r\n<p id=\"fs-id1340711\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(0,\u00b14\\right)\\,[\/latex]and foci[latex]\\,\\left(0,\u00b1\\sqrt{15}\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1376849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1376849\"]\r\n<p id=\"fs-id1376849\">[latex]{x}^{2}+\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1347532\" class=\"precalculus qa textbox shaded\">\r\n<p id=\"fs-id1347538\"><strong>Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?<\/strong><\/p>\r\n<p id=\"fs-id1363857\"><em>Yes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form[latex]\\,\\left(\u00b1a,0\\right)\\,[\/latex]or[latex]\\,\\left(0,\\,\u00b1a\\right).\\,[\/latex]Similarly, the coordinates of the foci will always have the form[latex]\\,\\left(\u00b1c,0\\right)\\,[\/latex]or[latex]\\,\\left(0,\\,\u00b1c\\right).\\,[\/latex]Knowing this, we can use[latex]\\,a\\,[\/latex]and[latex]\\,c\\,[\/latex]from the given points, along with the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2},[\/latex]to find[latex]\\,{b}^{2}.[\/latex]<\/em><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1316402\" class=\"bc-section section\">\r\n<h4>Writing Equations of Ellipses Not Centered at the Origin<\/h4>\r\n<p id=\"fs-id1316030\">Like the graphs of other equations, the graph of an <span class=\"no-emphasis\">ellipse<\/span> can be translated. If an ellipse is translated[latex]\\,h\\,[\/latex]units horizontally and[latex]\\,k\\,[\/latex]units vertically, the center of the ellipse will be[latex]\\,\\left(h,k\\right).\\,[\/latex]This <span class=\"no-emphasis\">translation<\/span> results in the standard form of the equation we saw previously, with[latex]\\,x\\,[\/latex]replaced by[latex]\\,\\left(x-h\\right)\\,[\/latex]and <em>y<\/em> replaced by[latex]\\,\\left(y-k\\right).[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1360258\" class=\"textbox key-takeaways\">\r\n<h3>Standard Forms of the Equation of an Ellipse with Center (<em>h<\/em>, <em>k<\/em>)<\/h3>\r\n<p id=\"fs-id1336434\">The standard form of the equation of an ellipse with center[latex]\\,\\left(h,\\text{ }k\\right)\\,[\/latex]and <span class=\"no-emphasis\">major axis<\/span> parallel to the <em>x<\/em>-axis is<\/p>\r\n\r\n<div id=\"Equation_10_01_03\">[latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/div>\r\n<p id=\"fs-id1355127\">where<\/p>\r\n\r\n<ul id=\"fs-id1355130\">\r\n \t<li>[latex]a&gt;b[\/latex]<\/li>\r\n \t<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(h\u00b1a,k\\right)[\/latex]<\/li>\r\n \t<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(h,k\u00b1b\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(h\u00b1c,k\\right),[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_006\">(Figure)<\/a><strong>a<\/strong><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1358861\">The standard form of the equation of an ellipse with center[latex]\\,\\left(h,k\\right)\\,[\/latex]and major axis parallel to the <em>y<\/em>-axis is<\/p>\r\n\r\n<div id=\"Equation_10_01_04\">[latex]\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1[\/latex]<\/div>\r\n<p id=\"fs-id1354473\">where<\/p>\r\n\r\n<ul id=\"fs-id1347416\">\r\n \t<li>[latex]a&gt;b[\/latex]<\/li>\r\n \t<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(h,k\u00b1a\\right)[\/latex]<\/li>\r\n \t<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(h\u00b1b,k\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(h,k\u00b1c\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_006\">(Figure)<\/a><strong>b<\/strong><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1307508\">Just as with ellipses centered at the origin, ellipses that are centered at a point[latex]\\,\\left(h,k\\right)\\,[\/latex]have vertices, co-vertices, and foci that are related by the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]We can use this relationship along with the midpoint and distance formulas to find the equation of the ellipse in standard form when the vertices and foci are given.<\/p>\r\n\r\n<div id=\"Figure_10_01_006\" class=\"wp-caption aligncenter\">\r\n<div class=\"wp-caption-text\"><\/div>\r\n[caption id=\"\" align=\"aligncenter\" width=\"935\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151241\/CNX_Precalc_Figure_10_01_006-1.jpg\" alt=\"\" width=\"935\" height=\"440\" \/> <strong>Figure 7. <\/strong>(a) Horizontal ellipse with center[latex]\\,\\left(h,k\\right)\\,[\/latex](b) Vertical ellipse with center[latex]\\,\\left(h,k\\right)[\/latex][\/caption]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1328854\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1328861\"><strong>Given the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1328866\" type=\"1\">\r\n \t<li>Determine whether the major axis is parallel to the <em>x<\/em>- or <em>y<\/em>-axis.\r\n<ol id=\"fs-id1332632\" type=\"a\">\r\n \t<li>If the <em>y<\/em>-coordinates of the given vertices and foci are the same, then the major axis is parallel to the <em>x<\/em>-axis. Use the standard form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1.[\/latex]<\/li>\r\n \t<li>If the <em>x<\/em>-coordinates of the given vertices and foci are the same, then the major axis is parallel to the <em>y<\/em>-axis. Use the standard form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1.[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Identify the center of the ellipse[latex]\\,\\left(h,k\\right)\\,[\/latex]using the midpoint formula and the given coordinates for the vertices.<\/li>\r\n \t<li>Find[latex]\\,{a}^{2}\\,[\/latex]by solving for the length of the major axis,[latex]\\,2a,[\/latex] which is the distance between the given vertices.<\/li>\r\n \t<li>Find[latex]\\,{c}^{2}\\,[\/latex]using[latex]\\,h\\,[\/latex]and[latex]\\,k,[\/latex] found in Step 2, along with the given coordinates for the foci.<\/li>\r\n \t<li>Solve for[latex]\\,{b}^{2}\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/li>\r\n \t<li>Substitute the values for[latex]\\,h,k,{a}^{2},[\/latex] and[latex]\\,{b}^{2}\\,[\/latex]into the standard form of the equation determined in Step 1.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_10_01_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1328619\">\r\n<div id=\"fs-id1328621\">\r\n<h3>Writing the Equation of an Ellipse Centered at a Point Other Than the Origin<\/h3>\r\n<p id=\"fs-id1328627\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(-2,-8\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{2}\\right)[\/latex]<\/p>\r\n<p id=\"fs-id1343972\">and foci[latex]\\,\\left(-2,-7\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{1}\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1342912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1342912\"]\r\n<p id=\"fs-id1342912\">The <em>x<\/em>-coordinates of the vertices and foci are the same, so the major axis is parallel to the <em>y<\/em>-axis. Thus, the equation of the ellipse will have the form<\/p>\r\n\r\n<div id=\"fs-id1355444\" class=\"unnumbered aligncenter\">[latex]\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1[\/latex]<\/div>\r\n<p id=\"fs-id1370586\">First, we identify the center,[latex]\\,\\left(h,k\\right).\\,[\/latex]The center is halfway between the vertices,[latex]\\,\\left(-2,-8\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{2}\\right).\\,[\/latex]Applying the midpoint formula, we have:<\/p>\r\n\r\n<div id=\"fs-id1372947\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\left(h,k\\right)=\\left(\\frac{-2+\\left(-2\\right)}{2},\\frac{-8+2}{2}\\right)\\hfill \\\\ \\text{ }=\\left(-2,-3\\right)\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1336766\">Next, we find[latex]\\,{a}^{2}.\\,[\/latex]The length of the major axis,[latex]\\,2a,[\/latex] is bounded by the vertices. We solve for[latex]\\,a\\,[\/latex]by finding the distance between the <em>y<\/em>-coordinates of the vertices.<\/p>\r\n\r\n<div id=\"fs-id1359290\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2a=2-\\left(-8\\right)\\\\ 2a=10\\\\ a=5\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1367878\">So[latex]\\,{a}^{2}=25.[\/latex]<\/p>\r\n<p id=\"fs-id1359360\">Now we find[latex]\\,{c}^{2}.\\,[\/latex]The foci are given by[latex]\\,\\left(h,k\u00b1c\\right).\\,[\/latex]So,[latex]\\,\\left(h,k-c\\right)=\\left(-2,-7\\right)\\,[\/latex]and[latex]\\,\\left(h,k+c\\right)=\\left(-2,\\text{1}\\right).\\,[\/latex]We substitute[latex]\\,k=-3\\,[\/latex]using either of these points to solve for[latex]\\,c.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1329404\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\,\\,\\,\\,k+c=1\\\\ -3+c=1\\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,c=4\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1368995\">So[latex]\\,{c}^{2}=16.[\/latex]<\/p>\r\n<p id=\"fs-id1336896\">Next, we solve for[latex]\\,{b}^{2}\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1346065\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{c}^{2}={a}^{2}-{b}^{2}\\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,16=25-{b}^{2}\\\\ {b}^{2}=9\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1385046\">Finally, we substitute the values found for[latex]\\,h,k,{a}^{2},[\/latex] and[latex]\\,{b}^{2}\\,[\/latex]into the standard form equation for an ellipse:<\/p>\r\n\r\n<div id=\"fs-id2289909\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{{\\left(x+2\\right)}^{2}}{9}+\\frac{{\\left(y+3\\right)}^{2}}{25}=1[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1346186\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_10_01_02\">\r\n<div id=\"fs-id1362367\">\r\n<p id=\"fs-id1362368\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(-3,3\\right)\\,[\/latex]and[latex]\\,\\left(5,3\\right)\\,[\/latex]and foci[latex]\\,\\left(1-2\\sqrt{3},3\\right)\\,[\/latex]and[latex]\\,\\left(1+2\\sqrt{3},3\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1385633\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1385633\"]\r\n<p id=\"fs-id1385633\">[latex]\\frac{{\\left(x-1\\right)}^{2}}{16}+\\frac{{\\left(y-3\\right)}^{2}}{4}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1291369\" class=\"bc-section section\">\r\n<h3>Graphing Ellipses Centered at the Origin<\/h3>\r\n<p id=\"fs-id1291374\">Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\text{ }a&gt;b\\,[\/latex]for horizontal ellipses and[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,\\text{ }a&gt;b\\,[\/latex]for vertical ellipses.<\/p>\r\n\r\n<div id=\"fs-id1333613\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1343715\"><strong>Given the standard form of an equation for an ellipse centered at[latex]\\,\\left(0,0\\right),[\/latex] sketch the graph.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1385565\" type=\"1\">\r\n \t<li>Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.\r\n<ol id=\"fs-id1333256\" type=\"a\">\r\n \t<li>If the equation is in the form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,a&gt;b,\\,[\/latex]then\r\n<ul id=\"fs-id1334467\">\r\n \t<li>the major axis is the <em>x<\/em>-axis<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(\u00b1a,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(0,\u00b1b\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(\u00b1c,0\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>If the equation is in the form[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,[\/latex]where[latex]\\,a&gt;b,\\,[\/latex]then\r\n<ul id=\"fs-id1333547\">\r\n \t<li>the major axis is the <em>y<\/em>-axis<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(0,\u00b1a\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(\u00b1b,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(0,\u00b1c\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Solve for[latex]\\,c\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/li>\r\n \t<li>Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_10_01_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1354398\">\r\n<div id=\"fs-id1346301\">\r\n<h3>Graphing an Ellipse Centered at the Origin<\/h3>\r\n<p id=\"fs-id1346306\">Graph the ellipse given by the equation,[latex]\\,\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{25}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1362387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1362387\"]\r\n<p id=\"fs-id1362387\">First, we determine the position of the major axis. Because[latex]\\,25&gt;9,[\/latex]the major axis is on the <em>y<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,[\/latex]where[latex]\\,{b}^{2}=9\\,[\/latex]and[latex]\\,{a}^{2}=25.\\,[\/latex]It follows that:<\/p>\r\n\r\n<ul id=\"fs-id1333413\">\r\n \t<li>the center of the ellipse is[latex]\\,\\left(0,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(0,\u00b1a\\right)=\\left(0,\u00b1\\sqrt{25}\\right)=\\left(0,\u00b15\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(\u00b1b,0\\right)=\\left(\u00b1\\sqrt{9},0\\right)=\\left(\u00b13,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(0,\u00b1c\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}\\,[\/latex]Solving for[latex]\\,c,[\/latex] we have:<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1328030\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{25-9}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{16}\\hfill \\\\ \\,\\,\\,=\u00b14\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1375192\">Therefore, the coordinates of the foci are[latex]\\,\\left(0,\u00b14\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1370885\">Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_007\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_10_01_007\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151248\/CNX_Precalc_Figure_10_01_007.jpg\" alt=\"\" width=\"731\" height=\"521\" \/> <strong>Figure 8.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1366513\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_10_01_03\">\r\n<div id=\"fs-id1341581\">\r\n<p id=\"fs-id1341582\">Graph the ellipse given by the equation[latex]\\,\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{4}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1355523\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1355523\"]\r\n<p id=\"fs-id1355523\">center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(\u00b16,0\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(0,\u00b12\\right);\\,[\/latex]foci:[latex]\\,\\left(\u00b14\\sqrt{2},0\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1354910\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151254\/CNX_Precalc_Figure_10_01_008.jpg\" alt=\"\" \/>[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_10_01_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1354926\">\r\n<div id=\"fs-id1354928\">\r\n<h3>Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form<\/h3>\r\n<p id=\"fs-id1354933\">Graph the ellipse given by the equation[latex]\\,4{x}^{2}+25{y}^{2}=100.\\,[\/latex]Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1343403\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1343403\"]\r\n<p id=\"fs-id1343403\">First, use algebra to rewrite the equation in standard form.<\/p>\r\n\r\n<div id=\"fs-id1372134\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l} 4{x}^{2}+25{y}^{2}=100\\hfill \\\\ \\text{ }\\frac{4{x}^{2}}{100}+\\frac{25{y}^{2}}{100}=\\frac{100}{100}\\hfill \\\\ \\text{ }\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1373251\">Next, we determine the position of the major axis. Because[latex]\\,25&gt;4,\\,[\/latex]the major axis is on the <em>x<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,{a}^{2}=25\\,[\/latex]and[latex]\\,{b}^{2}=4.\\,[\/latex]It follows that:<\/p>\r\n\r\n<ul id=\"fs-id1331280\">\r\n \t<li>the center of the ellipse is[latex]\\,\\left(0,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(\u00b1a,0\\right)=\\left(\u00b1\\sqrt{25},0\\right)=\\left(\u00b15,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(0,\u00b1b\\right)=\\left(0,\u00b1\\sqrt{4}\\right)=\\left(0,\u00b12\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(\u00b1c,0\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,\\,[\/latex]we have:<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1355541\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{25-4}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{21}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1377751\">Therefore the coordinates of the foci are[latex]\\,\\left(\u00b1\\sqrt{21},0\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1298485\">Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.<\/p>\r\n\r\n<div id=\"Figure_10_01_009\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151256\/CNX_Precalc_Figure_10_01_009.jpg\" alt=\"\" width=\"731\" height=\"366\" \/> <strong>Figure 9.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1298505\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_10_01_04\">\r\n<div id=\"fs-id1334387\">\r\n<p id=\"fs-id1334388\">Graph the ellipse given by the equation[latex]\\,49{x}^{2}+16{y}^{2}=784.\\,[\/latex]Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1229750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1229750\"]\r\n<p id=\"fs-id1229750\">Standard form:[latex]\\,\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{49}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(0,\u00b17\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(\u00b14,0\\right);\\,[\/latex]foci:[latex]\\,\\left(0,\u00b1\\sqrt{33}\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1316504\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151307\/CNX_Precalc_Figure_10_01_010.jpg\" alt=\"\" \/><\/span>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1316517\" class=\"bc-section section\">\r\n<h3>Graphing Ellipses Not Centered at the Origin<\/h3>\r\nWhen an <span class=\"no-emphasis\">ellipse<\/span> is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,[latex]\\,\\left(h,k\\right),[\/latex]we use the standard forms[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\text{ }a&gt;b\\,[\/latex]for horizontal ellipses and[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\text{ }a&gt;b\\,[\/latex]for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.\r\n<div id=\"fs-id1298882\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1298889\"><strong>Given the standard form of an equation for an ellipse centered at[latex]\\,\\left(h,k\\right),[\/latex] sketch the graph.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1358186\" type=\"1\">\r\n \t<li>Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.\r\n<ol id=\"fs-id1358193\" type=\"a\">\r\n \t<li>If the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,a&gt;b,\\,[\/latex]then\r\n<ul id=\"fs-id1367701\">\r\n \t<li>the center is[latex]\\,\\left(h,k\\right)[\/latex]<\/li>\r\n \t<li>the major axis is parallel to the <em>x<\/em>-axis<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(h\u00b1a,k\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(h,k\u00b1b\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(h\u00b1c,k\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>If the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\,[\/latex]where[latex]\\,a&gt;b,\\,[\/latex]then\r\n<ul id=\"fs-id1388657\">\r\n \t<li>the center is[latex]\\,\\left(h,k\\right)[\/latex]<\/li>\r\n \t<li>the major axis is parallel to the <em>y<\/em>-axis<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(h,k\u00b1a\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(h\u00b1b,k\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(h,k\u00b1c\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Solve for[latex]\\,c\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/li>\r\n \t<li>Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_10_01_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1358880\">\r\n<div id=\"fs-id1358883\">\r\n<h3>Graphing an Ellipse Centered at (<em>h<\/em>, <em>k<\/em>)<\/h3>\r\n<p id=\"fs-id1361566\">Graph the ellipse given by the equation,[latex]\\,\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y-5\\right)}^{2}}{9}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1351738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1351738\"]\r\n<p id=\"fs-id1351738\">First, we determine the position of the major axis. Because[latex]\\,9&gt;4,[\/latex] the major axis is parallel to the <em>y<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\,[\/latex]where[latex]\\,{b}^{2}=4\\,[\/latex]and[latex]\\,{a}^{2}=9.\\,[\/latex]It follows that:<\/p>\r\n\r\n<ul id=\"fs-id1290987\">\r\n \t<li>the center of the ellipse is[latex]\\,\\left(h,k\\right)=\\left(-2,\\text{5}\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(h,k\u00b1a\\right)=\\left(-2,5\u00b1\\sqrt{9}\\right)=\\left(-2,5\u00b13\\right),[\/latex] or[latex]\\,\\left(-2,\\text{2}\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{8}\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(h\u00b1b,k\\right)=\\left(-2\u00b1\\sqrt{4},5\\right)=\\left(-2\u00b12,5\\right),[\/latex] or[latex]\\,\\left(-4,5\\right)\\,[\/latex]and[latex]\\,\\left(0,\\text{5}\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(h,k\u00b1c\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,[\/latex]we have:<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1368094\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\end{array}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{9-4}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{5}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1194789\">Therefore, the coordinates of the foci are[latex]\\,\\left(-2,\\text{5}-\\sqrt{5}\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{5+}\\sqrt{5}\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1368273\">Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.<\/p>\r\n\r\n<div id=\"Figure_10_01_011\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151309\/CNX_Precalc_Figure_10_01_011.jpg\" alt=\"\" width=\"487\" height=\"441\" \/> <strong>Figure 10.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1316791\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_10_01_05\">\r\n<div id=\"fs-id1316801\">\r\n<p id=\"fs-id1316802\">Graph the ellipse given by the equation[latex]\\,\\frac{{\\left(x-4\\right)}^{2}}{36}+\\frac{{\\left(y-2\\right)}^{2}}{20}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1361229\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1361229\"]\r\n<p id=\"fs-id1361229\">Center:[latex]\\,\\left(4,2\\right);\\,[\/latex]vertices:[latex]\\,\\left(-2,2\\right)\\,[\/latex]and[latex]\\,\\left(10,2\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(4,2-2\\sqrt{5}\\right)\\,[\/latex]and[latex]\\,\\left(4,2+2\\sqrt{5}\\right);\\,[\/latex]foci:[latex]\\,\\left(0,2\\right)\\,[\/latex]and[latex]\\,\\left(8,2\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1365677\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151341\/CNX_Precalc_Figure_10_01_012.jpg\" alt=\"\" \/>[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1365689\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1365696\"><strong>Given the general form of an equation for an ellipse centered at (<em>h<\/em>, <em>k<\/em>), express the equation in standard form.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1366214\" type=\"1\">\r\n \t<li>Recognize that an ellipse described by an equation in the form[latex]\\,a{x}^{2}+b{y}^{2}+cx+dy+e=0\\,[\/latex]is in general form.<\/li>\r\n \t<li>Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.<\/li>\r\n \t<li>Factor out the coefficients of the[latex]\\,{x}^{2}\\,[\/latex]and[latex]\\,{y}^{2}\\,[\/latex]terms in preparation for completing the square.<\/li>\r\n \t<li>Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant,[latex]\\,{m}_{1}{\\left(x-h\\right)}^{2}+{m}_{2}{\\left(y-k\\right)}^{2}={m}_{3},[\/latex] where[latex]\\,{m}_{1},{m}_{2},[\/latex] and[latex]\\,{m}_{3}\\,[\/latex]are constants.<\/li>\r\n \t<li>Divide both sides of the equation by the constant term to express the equation in standard form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_10_01_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1353840\">\r\n<div id=\"fs-id1353842\">\r\n<h3>Graphing an Ellipse Centered at (<em>h<\/em>, <em>k<\/em>) by First Writing It in Standard Form<\/h3>\r\n<p id=\"fs-id1340159\">Graph the ellipse given by the equation[latex]\\,4{x}^{2}+9{y}^{2}-40x+36y+100=0.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1366619\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1366619\"]\r\n<p id=\"fs-id1366619\">We must begin by rewriting the equation in standard form.<\/p>\r\n\r\n<div id=\"fs-id1366623\" class=\"unnumbered aligncenter\">[latex]4{x}^{2}+9{y}^{2}-40x+36y+100=0[\/latex]<\/div>\r\n<p id=\"fs-id1367404\">Group terms that contain the same variable, and move the constant to the opposite side of the equation.<\/p>\r\n\r\n<div id=\"fs-id1367407\" class=\"unnumbered aligncenter\">[latex]\\left(4{x}^{2}-40x\\right)+\\left(9{y}^{2}+36y\\right)=-100[\/latex]<\/div>\r\n<p id=\"fs-id1350008\">Factor out the coefficients of the squared terms.<\/p>\r\n\r\n<div id=\"fs-id1350011\" class=\"unnumbered aligncenter\">[latex]4\\left({x}^{2}-10x\\right)+9\\left({y}^{2}+4y\\right)=-100[\/latex]<\/div>\r\n<p id=\"fs-id1313081\">Complete the square twice. Remember to balance the equation by adding the same constants to each side.<\/p>\r\n\r\n<div id=\"fs-id1313084\" class=\"unnumbered aligncenter\">[latex]4\\left({x}^{2}-10x+25\\right)+9\\left({y}^{2}+4y+4\\right)=-100+100+36[\/latex]<\/div>\r\n<p id=\"fs-id1360172\">Rewrite as perfect squares.<\/p>\r\n\r\n<div id=\"fs-id1360175\" class=\"unnumbered aligncenter\">[latex]4{\\left(x-5\\right)}^{2}+9{\\left(y+2\\right)}^{2}=36[\/latex]<\/div>\r\n<p id=\"fs-id1360100\">Divide both sides by the constant term to place the equation in standard form.<\/p>\r\n\r\n<div id=\"fs-id1360104\" class=\"unnumbered aligncenter\">[latex]\\frac{{\\left(x-5\\right)}^{2}}{9}+\\frac{{\\left(y+2\\right)}^{2}}{4}=1[\/latex]<\/div>\r\n<p id=\"fs-id1306079\">Now that the equation is in standard form, we can determine the position of the major axis. Because[latex]\\,9&gt;4,\\,[\/latex] the major axis is parallel to the <em>x<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,{a}^{2}=9\\,[\/latex]and[latex]\\,{b}^{2}=4.\\,[\/latex]It follows that:<\/p>\r\n\r\n<ul id=\"fs-id1336814\">\r\n \t<li>the center of the ellipse is[latex]\\,\\left(h,k\\right)=\\left(5,-2\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the vertices are[latex]\\,\\left(h\u00b1a,k\\right)=\\left(5\u00b1\\sqrt{9},-2\\right)=\\left(5\u00b13,-2\\right),\\,[\/latex]or[latex]\\,\\left(2,-2\\right)\\,[\/latex]and[latex]\\,\\left(8,-2\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the co-vertices are[latex]\\,\\left(h,k\u00b1b\\right)=\\left(\\text{5},-2\u00b1\\sqrt{4}\\right)=\\left(\\text{5},-2\u00b12\\right),\\,[\/latex]or[latex]\\,\\left(5,-4\\right)\\,[\/latex]and[latex]\\,\\left(5,\\text{0}\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are[latex]\\,\\left(h\u00b1c,k\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,\\,[\/latex]we have:<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1389238\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{9-4}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{5}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1355910\">Therefore, the coordinates of the foci are[latex]\\,\\left(\\text{5}-\\sqrt{5},-2\\right)\\,[\/latex]and[latex]\\,\\left(\\text{5+}\\sqrt{5},-2\\right).[\/latex]<\/p>\r\nNext we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_018\">(Figure)<\/a>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151347\/CNX_Precalc_Figure_10_01_018.jpg\" alt=\"\" width=\"487\" height=\"365\" \/> <strong>Figure 11.<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1396393\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_10_01_06\">\r\n<div id=\"fs-id1396402\">\r\n<p id=\"fs-id1396403\">Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.<\/p>\r\n\r\n<div id=\"fs-id1320958\" class=\"unnumbered aligncenter\">[latex]4{x}^{2}+{y}^{2}-24x+2y+21=0[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1388899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1388899\"]\r\n<p id=\"fs-id1388899\">[latex]\\,\\frac{{\\left(x-3\\right)}^{2}}{4}+\\frac{{\\left(y+1\\right)}^{2}}{16}=1;\\,[\/latex]center:[latex]\\,\\left(3,-1\\right);\\,[\/latex]vertices:[latex]\\,\\left(3,-\\text{5}\\right)\\,[\/latex]and[latex]\\,\\left(3,\\text{3}\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(1,-1\\right)\\,[\/latex]and[latex]\\,\\left(5,-1\\right);\\,[\/latex]foci:[latex]\\,\\left(3,-\\text{1}-2\\sqrt{3}\\right)\\,[\/latex]and[latex]\\,\\left(3,-\\text{1+}2\\sqrt{3}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1340629\" class=\"bc-section section\">\r\n<h3>Solving Applied Problems Involving Ellipses<\/h3>\r\n<p id=\"fs-id1340634\">Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_013\">(Figure)<\/a>. In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci\u2014about 43 feet apart\u2014can hear each other whisper.<\/p>\r\n\r\n<div id=\"Figure_10_01_013\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151357\/CNX_Precalc_Figure_10_01_013.jpg\" alt=\"\" width=\"487\" height=\"338\" \/> <strong>Figure 12.<\/strong>Sound waves are reflected between foci in an elliptical room, called a whispering chamber.[\/caption]\r\n\r\n<\/div>\r\n<div id=\"Example_10_01_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1377690\">\r\n<div id=\"fs-id1377692\">\r\n<h3>Locating the Foci of a Whispering Chamber<\/h3>\r\n<p id=\"fs-id1377697\">The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_015\">(Figure)<\/a>.<\/p>\r\n\r\n<ol id=\"fs-id1377705\" type=\"a\">\r\n \t<li>What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point[latex]\\,\\left(0,0\\right).[\/latex]<\/li>\r\n \t<li>If two senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.<\/li>\r\n<\/ol>\r\n<div id=\"Figure_10_01_015\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151401\/CNX_Precalc_Figure_10_01_015.jpg\" alt=\"\" width=\"487\" height=\"298\" \/> <strong>Figure 13.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1351971\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1351971\"]\r\n<ol id=\"fs-id1351971\" type=\"a\">\r\n \t<li>We are assuming a horizontal ellipse with center[latex]\\,\\left(0,0\\right),[\/latex] so we need to find an equation of the form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,a&gt;b.\\,[\/latex]We know that the length of the major axis,[latex]\\,2a,\\,[\/latex]is longer than the length of the minor axis,[latex]\\,2b.\\,[\/latex]So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis.\r\n<ul id=\"fs-id1361057\">\r\n \t<li>Solving for[latex]\\,a,[\/latex] we have[latex]\\,2a=96,[\/latex] so[latex]\\,a=48,[\/latex] and[latex]\\,{a}^{2}=2304.[\/latex]<\/li>\r\n \t<li>Solving for[latex]\\,b,[\/latex] we have[latex]\\,2b=46,[\/latex] so[latex]\\,b=23,[\/latex] and[latex]\\,{b}^{2}=529.[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1396675\">Therefore, the equation of the ellipse is[latex]\\,\\frac{{x}^{2}}{2304}+\\frac{{y}^{2}}{529}=1.[\/latex]<\/p>\r\n<\/li>\r\n \t<li>To find the distance between the senators, we must find the distance between the foci,[latex]\\,\\left(\u00b1c,0\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,[\/latex]we have:\r\n<div id=\"fs-id1363850\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}{c}^{2}={a}^{2}-{b}^{2}\\hfill &amp; \\hfill \\\\ {c}^{2}=2304-529\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Substitute using the values found in part (a)}.\\hfill \\\\ \\,\\,\\,c=\u00b1\\sqrt{2304-529}\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Take the square root of both sides}.\\hfill \\\\ \\,\\,\\,c=\u00b1\\sqrt{1775} \\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Subtract}.\\hfill \\\\ \\,\\,\\,c\\approx \u00b142\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Round to the nearest foot}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1372191\">The points[latex]\\,\\left(\u00b142,0\\right)\\,[\/latex]represent the foci. Thus, the distance between the senators is[latex]\\,2\\left(42\\right)=84\\,[\/latex]feet.[\/hidden-answer]<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1377180\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_10_01_07\">\r\n<div id=\"fs-id1377190\">\r\n<p id=\"fs-id1377191\">Suppose a whispering chamber is 480 feet long and 320 feet wide.<\/p>\r\n\r\n<ol id=\"fs-id1377194\" type=\"a\">\r\n \t<li>What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point[latex]\\,\\left(0,0\\right).[\/latex]<\/li>\r\n \t<li>If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1403959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1403959\"]\r\n<ol id=\"fs-id1403959\" type=\"a\">\r\n \t<li>[latex]\\frac{{x}^{2}}{57,600}+\\frac{{y}^{2}}{25,600}=1[\/latex]<\/li>\r\n \t<li>The people are standing 358 feet apart.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1315937\" class=\"precalculus media\">\r\n<p id=\"fs-id1315944\">Access these online resources for additional instruction and practice with ellipses.<\/p>\r\n\r\n<ul id=\"fs-id1315947\">\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/conicellipse\">Conic Sections: The Ellipse<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/grphellorigin\">Graph an Ellipse with Center at the Origin<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/grphellnot\">Graph an Ellipse with Center Not at the Origin<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1351649\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"fs-id1351656\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>Horizontal ellipse, center at origin<\/td>\r\n<td>[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Vertical ellipse, center at origin<\/td>\r\n<td>[latex]\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Horizontal ellipse, center [latex]\\,\\left(h,k\\right)[\/latex]<\/td>\r\n<td>[latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Vertical ellipse, center[latex]\\,\\left(h,k\\right)[\/latex]<\/td>\r\n<td>[latex]\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1368433\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1368439\">\r\n \t<li>An ellipse is the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).<\/li>\r\n \t<li>When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See <a class=\"autogenerated-content\" href=\"#Example_10_01_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_10_01_02\">(Figure)<\/a>.<\/li>\r\n \t<li>When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See <a class=\"autogenerated-content\" href=\"#Example_10_01_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_10_01_04\">(Figure)<\/a>.<\/li>\r\n \t<li>When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See <a class=\"autogenerated-content\" href=\"#Example_10_01_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_10_01_06\">(Figure)<\/a>.<\/li>\r\n \t<li>Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See <a class=\"autogenerated-content\" href=\"#Example_10_01_07\">(Figure)<\/a>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1343126\" class=\"textbox exercises\">\r\n<h3>Section Exercises<\/h3>\r\n<div id=\"fs-id1343129\" class=\"bc-section section\">\r\n<h4>Verbal<\/h4>\r\n<div id=\"fs-id1343134\">\r\n<div id=\"fs-id1343136\">\r\n<p id=\"fs-id1343137\">Define an ellipse in terms of its foci.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1374799\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1374799\"]\r\n<p id=\"fs-id1374799\">An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1374803\">\r\n<div id=\"fs-id1374804\">\r\n<p id=\"fs-id1374805\">Where must the foci of an ellipse lie?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1374809\">\r\n<div id=\"fs-id1374810\">\r\n<p id=\"fs-id1374811\">What special case of the ellipse do we have when the major and minor axis are of the same length?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1374815\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1374815\"]\r\n<p id=\"fs-id1374815\">This special case would be a circle.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1374818\">\r\n<div id=\"fs-id1374819\">\r\n<p id=\"fs-id1374820\">For the special case mentioned above, what would be true about the foci of that ellipse?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1374824\">\r\n<div id=\"fs-id1374825\">\r\n<p id=\"fs-id1374826\">What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the <em>y<\/em>-axis?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1374835\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1374835\"]\r\n<p id=\"fs-id1374835\">It is symmetric about the <em>x<\/em>-axis, <em>y<\/em>-axis, and the origin.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1351258\" class=\"bc-section section\">\r\n<h4>Algebraic<\/h4>\r\n<p id=\"fs-id1351264\">For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.<\/p>\r\n\r\n<div id=\"fs-id1351268\">\r\n<div id=\"fs-id1351269\">\r\n<p id=\"fs-id1351270\">[latex]2{x}^{2}+y=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1351300\">\r\n<div id=\"fs-id1351301\">\r\n<p id=\"fs-id1351302\">[latex]4{x}^{2}+9{y}^{2}=36[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1194849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1194849\"]\r\n<p id=\"fs-id1194849\">yes;[latex]\\,\\frac{{x}^{2}}{{3}^{2}}+\\frac{{y}^{2}}{{2}^{2}}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1339824\">\r\n<div id=\"fs-id1339826\">\r\n<p id=\"fs-id1339827\">[latex]4{x}^{2}-{y}^{2}=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1339864\">\r\n<div id=\"fs-id1339865\">\r\n<p id=\"fs-id1339866\">[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1377100\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1377100\"]\r\n<p id=\"fs-id1377100\">yes;[latex]\\frac{{x}^{2}}{{\\left(\\frac{1}{2}\\right)}^{2}}+\\frac{{y}^{2}}{{\\left(\\frac{1}{3}\\right)}^{2}}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1316571\">\r\n<div id=\"fs-id1316572\">\r\n<p id=\"fs-id1316573\">[latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1343327\">For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.<\/p>\r\n\r\n<div id=\"fs-id1350501\">\r\n<div id=\"fs-id1350502\">\r\n<p id=\"fs-id1350503\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{49}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1320315\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1320315\"]\r\n<p id=\"fs-id1320315\">[latex]\\frac{{x}^{2}}{{2}^{2}}+\\frac{{y}^{2}}{{7}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(0,7\\right)\\,[\/latex]and[latex]\\,\\left(0,-7\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(2,0\\right)\\,[\/latex]and[latex]\\,\\left(-2,0\\right).\\,[\/latex]Foci at[latex]\\,\\left(0,3\\sqrt{5}\\right),\\left(0,-3\\sqrt{5}\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1346718\">\r\n<div id=\"fs-id1346719\">\r\n<p id=\"fs-id1346720\">[latex]\\frac{{x}^{2}}{100}+\\frac{{y}^{2}}{64}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1359968\">\r\n<div id=\"fs-id1359969\">\r\n<p id=\"fs-id1359970\">[latex]{x}^{2}+9{y}^{2}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1385319\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1385319\"]\r\n<p id=\"fs-id1385319\">[latex]\\frac{{x}^{2}}{{\\left(1\\right)}^{2}}+\\frac{{y}^{2}}{{\\left(\\frac{1}{3}\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(1,0\\right)\\,[\/latex]and[latex]\\,\\left(-1,0\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(0,\\frac{1}{3}\\right),\\left(0,-\\frac{1}{3}\\right).\\,[\/latex]Foci at[latex]\\,\\left(\\frac{2\\sqrt{2}}{3},0\\right),\\left(-\\frac{2\\sqrt{2}}{3},0\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1369209\">\r\n<div id=\"fs-id1369210\">\r\n<p id=\"fs-id1369212\">[latex]4{x}^{2}+16{y}^{2}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1374887\">\r\n<div id=\"fs-id1374888\">\r\n<p id=\"fs-id1374889\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{49}+\\frac{{\\left(y-4\\right)}^{2}}{25}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1363641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1363641\"]\r\n<p id=\"fs-id1363641\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{{7}^{2}}+\\frac{{\\left(y-4\\right)}^{2}}{{5}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(9,4\\right),\\left(-5,4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(2,9\\right),\\left(2,-1\\right).\\,[\/latex]Foci at[latex]\\,\\left(2+2\\sqrt{6},4\\right),\\left(2-2\\sqrt{6},4\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1376228\">\r\n<div id=\"fs-id1376229\">\r\n<p id=\"fs-id1376230\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{81}+\\frac{{\\left(y+1\\right)}^{2}}{16}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1349361\">\r\n<div id=\"fs-id1349362\">\r\n<p id=\"fs-id1349363\">[latex]\\frac{{\\left(x+5\\right)}^{2}}{4}+\\frac{{\\left(y-7\\right)}^{2}}{9}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1359220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1359220\"]\r\n<p id=\"fs-id1359220\">[latex]\\frac{{\\left(x+5\\right)}^{2}}{{2}^{2}}+\\frac{{\\left(y-7\\right)}^{2}}{{3}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(-5,10\\right),\\left(-5,4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(-3,7\\right),\\left(-7,7\\right).\\,[\/latex]Foci at[latex]\\,\\left(-5,7+\\sqrt{5}\\right),\\left(-5,7-\\sqrt{5}\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1389365\">\r\n<div id=\"fs-id1389366\">\r\n<p id=\"fs-id1389367\">[latex]\\frac{{\\left(x-7\\right)}^{2}}{49}+\\frac{{\\left(y-7\\right)}^{2}}{49}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1340052\">\r\n<div id=\"fs-id1340053\">\r\n<p id=\"fs-id1340054\">[latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1365819\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1365819\"]\r\n<p id=\"fs-id1365819\">[latex]\\frac{{\\left(x-1\\right)}^{2}}{{3}^{2}}+\\frac{{\\left(y-4\\right)}^{2}}{{2}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(4,4\\right),\\left(-2,4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(1,6\\right),\\left(1,2\\right).\\,[\/latex]Foci at[latex]\\,\\left(1+\\sqrt{5},4\\right),\\left(1-\\sqrt{5},4\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1326905\">\r\n<div id=\"fs-id1326906\">\r\n<p id=\"fs-id1326907\">[latex]9{x}^{2}-54x+9{y}^{2}-54y+81=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1371116\">\r\n<div id=\"fs-id1371117\">\r\n<p id=\"fs-id1371118\">[latex]4{x}^{2}-24x+36{y}^{2}-360y+864=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1342974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1342974\"]\r\n<p id=\"fs-id1342974\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{{\\left(3\\sqrt{2}\\right)}^{2}}+\\frac{{\\left(y-5\\right)}^{2}}{{\\left(\\sqrt{2}\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(3+3\\sqrt{2},5\\right),\\left(3-3\\sqrt{2},5\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(3,5+\\sqrt{2}\\right),\\left(3,5-\\sqrt{2}\\right).\\,[\/latex]Foci at[latex]\\,\\left(7,5\\right),\\left(-1,5\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id693861\">\r\n<div id=\"fs-id693862\">\r\n<p id=\"fs-id693863\">[latex]4{x}^{2}+24x+16{y}^{2}-128y+228=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1364961\">\r\n<div id=\"fs-id1364962\">\r\n<p id=\"fs-id1364963\">[latex]4{x}^{2}+40x+25{y}^{2}-100y+100=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1361188\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1361188\"]\r\n<p id=\"fs-id1361188\">[latex]\\frac{{\\left(x+5\\right)}^{2}}{{\\left(5\\right)}^{2}}+\\frac{{\\left(y-2\\right)}^{2}}{{\\left(2\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(0,2\\right),\\left(-10,2\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(-5,4\\right),\\left(-5,0\\right).\\,[\/latex]Foci at[latex]\\,\\left(-5+\\sqrt{21},2\\right),\\left(-5-\\sqrt{21},2\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1332177\">\r\n<div id=\"fs-id1332178\">\r\n<p id=\"fs-id1332179\">[latex]{x}^{2}+2x+100{y}^{2}-1000y+2401=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1313180\">\r\n<div id=\"fs-id1313181\">\r\n<p id=\"fs-id1313182\">[latex]4{x}^{2}+24x+25{y}^{2}+200y+336=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"1332424\"]Show Solution[\/reveal-answer][hidden-answer a=\"1332424\"]\r\n\r\n[latex]\\frac{{\\left(x+3\\right)}^{2}}{{\\left(5\\right)}^{2}}+\\frac{{\\left(y+4\\right)}^{2}}{{\\left(2\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(2,-4\\right),\\left(-8,-4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(-3,-2\\right),\\left(-3,-6\\right).\\,[\/latex]Foci at[latex]\\,\\left(-3+\\sqrt{21},-4\\right),\\left(-3-\\sqrt{21},-4\\right).[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1327267\">\r\n<div id=\"fs-id1327268\">\r\n<p id=\"fs-id1327270\">[latex]9{x}^{2}+72x+16{y}^{2}+16y+4=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1332231\">For the following exercises, find the foci for the given ellipses.<\/p>\r\n\r\n<div id=\"fs-id1332234\">\r\n<div id=\"fs-id1332235\">\r\n<p id=\"fs-id1332236\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{25}+\\frac{{\\left(y+1\\right)}^{2}}{36}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1332043\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1332043\"]\r\n<p id=\"fs-id1332043\">Foci[latex]\\,\\left(-3,-1+\\sqrt{11}\\right),\\left(-3,-1-\\sqrt{11}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1313640\">\r\n<div id=\"fs-id1313641\">\r\n<p id=\"fs-id1313642\">[latex]\\frac{{\\left(x+1\\right)}^{2}}{100}+\\frac{{\\left(y-2\\right)}^{2}}{4}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1396602\">\r\n<div id=\"fs-id1396603\">\r\n<p id=\"fs-id1396604\">[latex]{x}^{2}+{y}^{2}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1396641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1396641\"]\r\n<p id=\"fs-id1396641\">Focus[latex]\\,\\left(0,0\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1328074\">\r\n<div id=\"fs-id1328075\">\r\n<p id=\"fs-id1328076\">[latex]{x}^{2}+4{y}^{2}+4x+8y=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1328126\">\r\n<div id=\"fs-id1328127\">\r\n<p id=\"fs-id1328128\">[latex]10{x}^{2}+{y}^{2}+200x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1346558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1346558\"]\r\n<p id=\"fs-id1346558\">Foci[latex]\\,\\left(-10,30\\right),\\left(-10,-30\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1377000\" class=\"bc-section section\">\r\n<h4>Graphical<\/h4>\r\n<p id=\"fs-id1377005\">For the following exercises, graph the given ellipses, noting center, vertices, and foci.<\/p>\r\n\r\n<div id=\"fs-id1377008\">\r\n<div id=\"fs-id1377009\">\r\n<p id=\"fs-id1377010\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{36}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1331349\">\r\n<div id=\"fs-id1331350\">\r\n<p id=\"fs-id1331351\">[latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1368187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1368187\"]\r\n<p id=\"fs-id1368187\">Center[latex]\\,\\left(0,0\\right),\\,[\/latex]Vertices[latex]\\,\\left(4,0\\right),\\left(-4,0\\right),\\left(0,3\\right),\\left(0,-3\\right),\\,[\/latex]Foci[latex]\\,\\left(\\sqrt{7},0\\right),\\left(-\\sqrt{7},0\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1360065\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151404\/CNX_Precalc_Figure_10_01_202.jpg\" alt=\"\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1360074\">\r\n<div id=\"fs-id1360075\">\r\n<p id=\"fs-id1360076\">[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1404169\">\r\n<div id=\"fs-id1404170\">\r\n<p id=\"fs-id1404171\">[latex]81{x}^{2}+49{y}^{2}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1404211\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1404211\"]\r\n<p id=\"fs-id1404211\">Center[latex]\\,\\left(0,0\\right),\\,[\/latex]Vertices[latex]\\,\\left(\\frac{1}{9},0\\right),\\left(-\\frac{1}{9},0\\right),\\left(0,\\frac{1}{7}\\right),\\left(0,-\\frac{1}{7}\\right),\\,[\/latex]Foci[latex]\\,\\left(0,\\frac{4\\sqrt{2}}{63}\\right),\\left(0,-\\frac{4\\sqrt{2}}{63}\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1329173\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151419\/CNX_Precalc_Figure_10_01_204.jpg\" alt=\"\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1329183\">\r\n<div id=\"fs-id1329184\">\r\n<p id=\"fs-id1329185\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{64}+\\frac{{\\left(y-4\\right)}^{2}}{16}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1326325\">\r\n<div id=\"fs-id1326326\">\r\n<p id=\"fs-id1326327\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{9}+\\frac{{\\left(y-3\\right)}^{2}}{9}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1388768\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1388768\"]\r\n<p id=\"fs-id1388768\">Center[latex]\\,\\left(-3,3\\right),\\,[\/latex]Vertices[latex]\\,\\left(0,3\\right),\\left(-6,3\\right),\\left(-3,0\\right),\\left(-3,6\\right),\\,[\/latex]Focus[latex]\\,\\left(-3,3\\right)\\,[\/latex]<\/p>\r\n<p id=\"fs-id1327102\">Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.<\/p>\r\n<span id=\"fs-id1327108\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151434\/CNX_Precalc_Figure_10_01_206.jpg\" alt=\"\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1354710\">\r\n<div id=\"fs-id1354711\">\r\n<p id=\"fs-id1354712\">[latex]\\frac{{x}^{2}}{2}+\\frac{{\\left(y+1\\right)}^{2}}{5}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1332813\">\r\n<div id=\"fs-id1332814\">\r\n<p id=\"fs-id1332815\">[latex]4{x}^{2}-8x+16{y}^{2}-32y-44=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1332872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1332872\"]\r\n<p id=\"fs-id1332872\">Center[latex]\\,\\left(1,1\\right),\\,[\/latex]Vertices[latex]\\,\\left(5,1\\right),\\left(-3,1\\right),\\left(1,3\\right),\\left(1,-1\\right),\\,[\/latex]Foci[latex]\\,\\left(1,1+4\\sqrt{3}\\right),\\left(1,1-4\\sqrt{3}\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1363950\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151437\/CNX_Precalc_Figure_10_01_208.jpg\" alt=\"\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1363960\">\r\n<div id=\"fs-id1363961\">\r\n<p id=\"fs-id1363962\">[latex]{x}^{2}-8x+25{y}^{2}-100y+91=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1326410\">\r\n<div id=\"fs-id1326411\">\r\n<p id=\"fs-id1326412\">[latex]{x}^{2}+8x+4{y}^{2}-40y+112=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1326468\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1326468\"]\r\n<p id=\"fs-id1326468\">Center[latex]\\,\\left(-4,5\\right),\\,[\/latex]Vertices[latex]\\,\\left(-2,5\\right),\\left(-6,4\\right),\\left(-4,6\\right),\\left(-4,4\\right),\\,[\/latex]Foci[latex]\\,\\left(-4+\\sqrt{3},5\\right),\\left(-4-\\sqrt{3},5\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1344024\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151443\/CNX_Precalc_Figure_10_01_210.jpg\" alt=\"\" \/>[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1344033\">\r\n<div id=\"fs-id1344034\">\r\n<p id=\"fs-id1344036\">[latex]64{x}^{2}+128x+9{y}^{2}-72y-368=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1333303\">\r\n<div id=\"fs-id1333304\">\r\n<p id=\"fs-id1333305\">[latex]16{x}^{2}+64x+4{y}^{2}-8y+4=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1194702\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1194702\"]\r\n<p id=\"fs-id1194702\">Center[latex]\\,\\left(-2,1\\right),\\,[\/latex]Vertices[latex]\\,\\left(0,1\\right),\\left(-4,1\\right),\\left(-2,5\\right),\\left(-2,-3\\right),\\,[\/latex]Foci[latex]\\,\\left(-2,1+2\\sqrt{3}\\right),\\left(-2,1-2\\sqrt{3}\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1396328\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151449\/CNX_Precalc_Figure_10_01_212.jpg\" alt=\"\" \/>[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1396338\">\r\n<div id=\"fs-id1396339\">\r\n<p id=\"fs-id1396340\">[latex]100{x}^{2}+1000x+{y}^{2}-10y+2425=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1363717\">\r\n<div id=\"fs-id1363718\">\r\n<p id=\"fs-id1363719\">[latex]4{x}^{2}+16x+4{y}^{2}+16y+16=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1326715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1326715\"]\r\n<p id=\"fs-id1326715\">Center[latex]\\,\\left(-2,-2\\right),\\,[\/latex]Vertices[latex]\\,\\left(0,-2\\right),\\left(-4,-2\\right),\\left(-2,0\\right),\\left(-2,-4\\right),\\,[\/latex]Focus[latex]\\,\\left(-2,-2\\right)[\/latex]<\/p>\r\n<span id=\"fs-id1327616\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151451\/CNX_Precalc_Figure_10_01_214.jpg\" alt=\"\" \/>[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1327626\">For the following exercises, use the given information about the graph of each ellipse to determine its equation.<\/p>\r\n\r\n<div id=\"fs-id1327630\">\r\n<div id=\"fs-id1327631\">\r\n<p id=\"fs-id1327632\">Center at the origin, symmetric with respect to the <em>x<\/em>- and <em>y<\/em>-axes, focus at[latex]\\,\\left(4,0\\right),[\/latex] and point on graph[latex]\\,\\left(0,3\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1304004\">\r\n<div id=\"fs-id1304005\">\r\n<p id=\"fs-id1304006\">Center at the origin, symmetric with respect to the <em>x<\/em>- and <em>y<\/em>-axes, focus at[latex]\\,\\left(0,-2\\right),[\/latex] and point on graph[latex]\\,\\left(5,0\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1368635\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1368635\"]\r\n<p id=\"fs-id1368635\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{29}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1368697\">\r\n<div id=\"fs-id1368698\">\r\n<p id=\"fs-id1368699\">Center at the origin, symmetric with respect to the <em>x<\/em>- and <em>y<\/em>-axes, focus at[latex]\\,\\left(3,0\\right),[\/latex] and major axis is twice as long as minor axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1320506\">\r\n<div id=\"fs-id1320507\">\r\n<p id=\"fs-id1320508\">Center[latex]\\,\\left(4,2\\right)[\/latex]; vertex[latex]\\,\\left(9,2\\right)[\/latex]; one focus:[latex]\\,\\left(4+2\\sqrt{6},2\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1364314\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1364314\"]\r\n<p id=\"fs-id1364314\">[latex]\\frac{{\\left(x-4\\right)}^{2}}{25}+\\frac{{\\left(y-2\\right)}^{2}}{1}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1303537\">\r\n<div id=\"fs-id1303538\">\r\n<p id=\"fs-id1303539\">Center[latex]\\,\\left(3,5\\right)[\/latex]; vertex[latex]\\,\\left(3,11\\right)[\/latex]; one focus:[latex]\\,\\left(3,\\text{ 5+4}\\sqrt{\\text{2}}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1346675\">\r\n<div id=\"fs-id1346676\">\r\n<p id=\"fs-id1346678\">Center[latex]\\,\\left(-3,4\\right)[\/latex]; vertex[latex]\\,\\left(1,4\\right)[\/latex]; one focus:[latex]\\,\\left(-3+2\\sqrt{3},4\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1396512\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1396512\"]\r\n<p id=\"fs-id1396512\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{16}+\\frac{{\\left(y-4\\right)}^{2}}{4}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1343632\">For the following exercises, given the graph of the ellipse, determine its equation.<\/p>\r\n\r\n<div id=\"fs-id1343635\">\r\n<div id=\"fs-id1343636\"><span id=\"fs-id1319617\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151500\/CNX_Precalc_Figure_10_01_215.jpg\" alt=\"\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1319627\">\r\n<div id=\"fs-id1319628\"><span id=\"fs-id1319633\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151513\/CNX_Precalc_Figure_10_01_216.jpg\" alt=\"\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1319644\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1319644\"]\r\n<p id=\"fs-id1319644\">[latex]\\frac{{x}^{2}}{81}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1349436\">\r\n<div id=\"fs-id1349437\"><span id=\"fs-id1349442\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151515\/CNX_Precalc_Figure_10_01_217.jpg\" alt=\"\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1349453\">\r\n<div id=\"fs-id1349454\"><span id=\"fs-id1349459\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151517\/CNX_Precalc_Figure_10_01_218.jpg\" alt=\"\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1349470\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1349470\"]\r\n<p id=\"fs-id1349470\">[latex]\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y-2\\right)}^{2}}{9}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1402281\">\r\n<div id=\"fs-id1402282\"><span id=\"fs-id1402287\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151523\/CNX_Precalc_Figure_10_01_219.jpg\" alt=\"\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1402299\" class=\"bc-section section\">\r\n<h4>Extensions<\/h4>\r\n<p id=\"fs-id1402304\">For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula[latex]\\,\\text{Area}=a\\cdot b\\cdot \\pi .[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1345055\">\r\n<div id=\"fs-id1345056\">\r\n<p id=\"fs-id1345058\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{9}+\\frac{{\\left(y-3\\right)}^{2}}{16}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1365738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1365738\"]\r\n<p id=\"fs-id1365738\">[latex]\\text{Area = 12\u03c0}\\,\\text{square}\\,\\text{units}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1365758\">\r\n<div id=\"fs-id1365759\">\r\n<p id=\"fs-id1365760\">[latex]\\frac{{\\left(x+6\\right)}^{2}}{16}+\\frac{{\\left(y-6\\right)}^{2}}{36}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1359643\">\r\n<div id=\"fs-id1359644\">\r\n<p id=\"fs-id1359646\">[latex]\\frac{{\\left(x+1\\right)}^{2}}{4}+\\frac{{\\left(y-2\\right)}^{2}}{5}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1343831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1343831\"]\r\n<p id=\"fs-id1343831\">[latex]\\text{Area = 2}\\sqrt{\\text{5}}\\text{\u03c0}\\,\\text{square}\\,\\text{units}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1343860\">\r\n<div id=\"fs-id1343861\">\r\n<p id=\"fs-id1319526\">[latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1319582\">\r\n<div id=\"fs-id1319583\">\r\n<p id=\"fs-id1319584\">[latex]9{x}^{2}-54x+9{y}^{2}-54y+81=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1404247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1404247\"]\r\n<p id=\"fs-id1404247\">[latex]\\text{Area = 9\u03c0}\\,\\text{square}\\,\\text{units}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1404269\" class=\"bc-section section\">\r\n<h4>Real-World Applications<\/h4>\r\n<div>\r\n<div id=\"fs-id1404275\">\r\n<p id=\"fs-id1404276\">Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1404280\">\r\n<div id=\"fs-id1404282\">\r\n<p id=\"fs-id1404283\">Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of[latex]\\,h,[\/latex] the height.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1404304\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1404304\"]\r\n<p id=\"fs-id1404304\">[latex]\\frac{{x}^{2}}{4{h}^{2}}+\\frac{{y}^{2}}{\\frac{1}{4}{h}^{2}}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1377342\">\r\n<div id=\"fs-id1377343\">\r\n<p id=\"fs-id1377344\">An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of the arch at a distance of 4 feet from the center.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1377350\">\r\n<div id=\"fs-id1377351\">\r\n<p id=\"fs-id1377352\">An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1352044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1352044\"]\r\n<p id=\"fs-id1352044\">[latex]\\frac{{x}^{2}}{400}+\\frac{{y}^{2}}{144}=1[\/latex]. Distance = 17.32 feet<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1352104\">\r\n<div id=\"fs-id1352106\">\r\n<p id=\"fs-id1352107\">A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1352112\">\r\n<div id=\"fs-id1352113\">\r\n<p id=\"fs-id1352114\">A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1352121\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1352121\"]\r\n<p id=\"fs-id1352121\">Approximately 51.96 feet<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1352125\">\r\n<div id=\"fs-id1352126\">\r\n<p id=\"fs-id1352127\">A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and height at the center of the gallery?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1352137\">\r\n \t<dt>center of an ellipse<\/dt>\r\n \t<dd id=\"fs-id1345665\">the midpoint of both the major and minor axes<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1345668\">\r\n \t<dt>conic section<\/dt>\r\n \t<dd id=\"fs-id1345672\">any shape resulting from the intersection of a right circular cone with a plane<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1345675\">\r\n \t<dt>ellipse<\/dt>\r\n \t<dd id=\"fs-id1345680\">the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]in a plane such that the sum of their distances from two fixed points is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1345712\">\r\n \t<dt>foci<\/dt>\r\n \t<dd id=\"fs-id1345716\">plural of focus<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1345719\">\r\n \t<dt>focus (of an ellipse)<\/dt>\r\n \t<dd id=\"fs-id1345723\">one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point[latex]\\,\\left(x,y\\right)\\,[\/latex]on the ellipse is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1345756\">\r\n \t<dt>major axis<\/dt>\r\n \t<dd id=\"fs-id1350559\">the longer of the two axes of an ellipse<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1350562\">\r\n \t<dt>minor axis<\/dt>\r\n \t<dd id=\"fs-id1350566\">the shorter of the two axes of an ellipse<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Write equations of ellipses in standard form.<\/li>\n<li>Graph ellipses centered at the origin.<\/li>\n<li>Graph ellipses not centered at the origin.<\/li>\n<li>Solve applied problems involving ellipses.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_10_01_001\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151211\/CNX_Precalc_Figure_10_01_001n.jpg\" alt=\"\" width=\"488\" height=\"324\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1. <\/strong>The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id693728\">Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_001\">(Figure)<\/a>, is such a room.<a class=\"footnote\" title=\"Architect of the Capitol. http:\/\/www.aoc.gov. Accessed April 15, 2014.\" id=\"return-footnote-3229-1\" href=\"#footnote-3229-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> It is an oval-shaped room called a <em>whispering chamber<\/em> because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.<\/p>\n<div id=\"fs-id1297573\" class=\"bc-section section\">\n<h3>Writing Equations of Ellipses in Standard Form<\/h3>\n<p id=\"fs-id1332910\">A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_002\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_01_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 986px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151214\/CNX_Precalc_Figure_10_01_002.jpg\" alt=\"\" width=\"976\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1332535\">Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).<\/p>\n<p id=\"fs-id1133644\">We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_01_003\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151217\/CNX_Precalc_Figure_10_01_003.jpg\" alt=\"\" width=\"487\" height=\"560\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1364494\">Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_004\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_01_004\" class=\"medium\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151227\/CNX_Precalc_Figure_10_01_004.jpg\" alt=\"\" width=\"731\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1337244\">In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the <em>x<\/em>&#8211; and <em>y<\/em>-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.<\/p>\n<p id=\"fs-id1194638\">To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.<\/p>\n<div id=\"fs-id1366768\" class=\"bc-section section\">\n<h4>Deriving the Equation of an Ellipse Centered at the Origin<\/h4>\n<p id=\"fs-id1297787\">To derive the equation of an <span class=\"no-emphasis\">ellipse<\/span> centered at the origin, we begin with the foci[latex]\\,\\left(-c,0\\right)\\,[\/latex]and[latex]\\,\\left(c,0\\right).\\,[\/latex]The ellipse is the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]such that the sum of the distances from[latex]\\,\\left(x,y\\right)\\,[\/latex]to the foci is constant, as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_014\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_01_014\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151234\/CNX_Precalc_Figure_10_01_014.jpg\" alt=\"\" width=\"487\" height=\"274\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1226529\">If[latex]\\,\\left(a,0\\right)\\,[\/latex]is a <span class=\"no-emphasis\">vertex<\/span> of the ellipse, the distance from[latex]\\,\\left(-c,0\\right)\\,[\/latex]to[latex]\\,\\left(a,0\\right)\\,[\/latex]is[latex]\\,a-\\left(-c\\right)=a+c.\\,[\/latex]The distance from[latex]\\,\\left(c,0\\right)\\,[\/latex]to[latex]\\,\\left(a,0\\right)\\,[\/latex]is[latex]\\,a-c[\/latex]. The sum of the distances from the <span class=\"no-emphasis\">foci<\/span> to the vertex is<\/p>\n<div id=\"fs-id1162201\" class=\"unnumbered aligncenter\">[latex]\\left(a+c\\right)+\\left(a-c\\right)=2a[\/latex]<\/div>\n<p id=\"fs-id1347565\">If[latex]\\,\\left(x,y\\right)\\,[\/latex]is a point on the ellipse, then we can define the following variables:<\/p>\n<div id=\"fs-id1105711\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{d}_{1}=\\text{the distance from }\\left(-c,0\\right)\\text{ to }\\left(x,y\\right)\\hfill \\\\ {d}_{2}=\\text{the distance from }\\left(c,0\\right)\\text{ to }\\left(x,y\\right)\\hfill \\end{array}[\/latex]<\/div>\n<p>By the definition of an ellipse,[latex]\\,{d}_{1}+{d}_{2}\\,[\/latex]is constant for any point[latex]\\,\\left(x,y\\right)\\,[\/latex]on the ellipse. We know that the sum of these distances is[latex]\\,2a\\,[\/latex]for the vertex[latex]\\,\\left(a,0\\right).\\,[\/latex]It follows that[latex]\\,{d}_{1}+{d}_{2}=2a\\,[\/latex]for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.<\/p>\n<div id=\"fs-id1106644\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }{d}_{1}+{d}_{2}=\\sqrt{{\\left(x-\\left(-c\\right)\\right)}^{2}+{\\left(y-0\\right)}^{2}}+\\sqrt{{\\left(x-c\\right)}^{2}+{\\left(y-0\\right)}^{2}}=2a\\hfill & \\text{Distance formula}\\hfill \\\\ \\sqrt{{\\left(x+c\\right)}^{2}+{y}^{2}}+\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}=2a\\hfill & \\text{Simplify expressions}\\text{.}\\hfill \\\\ \\text{ }\\sqrt{{\\left(x+c\\right)}^{2}+{y}^{2}}=2a-\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\hfill & \\text{Move radical to opposite side}\\text{.}\\hfill \\\\ \\text{ }{\\left(x+c\\right)}^{2}+{y}^{2}={\\left[2a-\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\right]}^{2}\\hfill & \\text{Square both sides}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}+2cx+{c}^{2}+{y}^{2}=4{a}^{2}-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}+{\\left(x-c\\right)}^{2}+{y}^{2}\\hfill & \\text{Expand the squares}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}+2cx+{c}^{2}+{y}^{2}=4{a}^{2}-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}+{x}^{2}-2cx+{c}^{2}+{y}^{2}\\hfill & \\text{Expand remaining squares}\\text{.}\\hfill \\\\ \\text{ }2cx=4{a}^{2}-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}-2cx\\hfill & \\text{Combine like terms}\\text{.}\\hfill \\\\ \\text{ }4cx-4{a}^{2}=-4a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\hfill & \\text{Isolate the radical}\\text{.}\\hfill \\\\ \\text{ }cx-{a}^{2}=-a\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\hfill & \\text{Divide by 4}\\text{.}\\hfill \\\\ \\text{ }{\\left[cx-{a}^{2}\\right]}^{2}={a}^{2}{\\left[\\sqrt{{\\left(x-c\\right)}^{2}+{y}^{2}}\\right]}^{2}\\hfill & \\text{Square both sides}\\text{.}\\hfill \\\\ \\text{ }{c}^{2}{x}^{2}-2{a}^{2}cx+{a}^{4}={a}^{2}\\left({x}^{2}-2cx+{c}^{2}+{y}^{2}\\right)\\hfill & \\text{Expand the squares}\\text{.}\\hfill \\\\ \\text{ }{c}^{2}{x}^{2}-2{a}^{2}cx+{a}^{4}={a}^{2}{x}^{2}-2{a}^{2}cx+{a}^{2}{c}^{2}+{a}^{2}{y}^{2}\\hfill & \\text{Distribute }{a}^{2}.\\hfill \\\\ \\text{ }{a}^{2}{x}^{2}-{c}^{2}{x}^{2}+{a}^{2}{y}^{2}={a}^{4}-{a}^{2}{c}^{2}\\hfill & \\text{Rewrite}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}\\left({a}^{2}-{c}^{2}\\right)+{a}^{2}{y}^{2}={a}^{2}\\left({a}^{2}-{c}^{2}\\right)\\hfill & \\text{Factor common terms}\\text{.}\\hfill \\\\ \\text{ }{x}^{2}{b}^{2}+{a}^{2}{y}^{2}={a}^{2}{b}^{2}\\hfill & \\text{Set }{b}^{2}={a}^{2}-{c}^{2}.\\hfill \\\\ \\text{ }\\frac{{x}^{2}{b}^{2}}{{a}^{2}{b}^{2}}+\\frac{{a}^{2}{y}^{2}}{{a}^{2}{b}^{2}}=\\frac{{a}^{2}{b}^{2}}{{a}^{2}{b}^{2}}\\hfill & \\text{Divide both sides by }{a}^{2}{b}^{2}.\\hfill \\\\ \\text{ }\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\hfill & \\text{Simplify}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1373013\">Thus, the standard equation of an ellipse is[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1.[\/latex]This equation defines an ellipse centered at the origin. If[latex]\\,a>b,[\/latex]the ellipse is stretched further in the horizontal direction, and if[latex]\\,b>a,[\/latex] the ellipse is stretched further in the vertical direction.<\/p>\n<\/div>\n<div id=\"fs-id1381696\" class=\"bc-section section\">\n<h4>Writing Equations of Ellipses Centered at the Origin in Standard Form<\/h4>\n<p id=\"fs-id1376358\">Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we\u2019ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.<\/p>\n<p id=\"fs-id1308221\">The key features of the <span class=\"no-emphasis\">ellipse<\/span> are its center, <span class=\"no-emphasis\">vertices<\/span>, <span class=\"no-emphasis\">co-vertices<\/span>, <span class=\"no-emphasis\">foci<\/span>, and lengths and positions of the <span class=\"no-emphasis\">major and minor axes<\/span>. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.<\/p>\n<div id=\"fs-id1343526\" class=\"textbox key-takeaways\">\n<h3>Standard Forms of the Equation of an Ellipse with Center (0,0)<\/h3>\n<p id=\"fs-id1389856\">The standard form of the equation of an ellipse with center[latex]\\,\\left(0,0\\right)\\,[\/latex]and major axis on the <em>x-axis<\/em> is<\/p>\n<div id=\"Equation_10_01_01\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/div>\n<p id=\"fs-id1321099\">where<\/p>\n<ul id=\"fs-id1321102\">\n<li>[latex]a>b[\/latex]<\/li>\n<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(\u00b1a,0\\right)[\/latex]<\/li>\n<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(0,\u00b1b\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(\u00b1c,0\\right)[\/latex], where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_005\">(Figure)<\/a><strong>a<\/strong><\/li>\n<\/ul>\n<p id=\"fs-id1327739\">The standard form of the equation of an ellipse with center[latex]\\,\\left(0,0\\right)\\,[\/latex]and major axis on the <em>y-axis<\/em> is<\/p>\n<div id=\"Equation_10_01_02\">[latex]\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1[\/latex]<\/div>\n<p id=\"fs-id1390177\">where<\/p>\n<ul id=\"fs-id1390180\">\n<li>[latex]a>b[\/latex]<\/li>\n<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(0,\u00b1a\\right)[\/latex]<\/li>\n<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(\u00b1b,0\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(0,\u00b1c\\right)[\/latex], where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_005\">(Figure)<\/a><strong>b<\/strong><\/li>\n<\/ul>\n<p id=\"fs-id1333061\">Note that the vertices, co-vertices, and foci are related by the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.<\/p>\n<div id=\"Figure_10_01_005\" class=\"wp-caption aligncenter\">\n<div style=\"width: 945px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151236\/CNX_Precalc_Figure_10_01_005-1.jpg\" alt=\"\" width=\"935\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6. <\/strong>(a) Horizontal ellipse with center[latex]\\,\\left(0,0\\right)\\,[\/latex](b) Vertical ellipse with center[latex]\\,\\left(0,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1369821\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1278660\"><strong>Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.<\/strong><\/p>\n<ol id=\"fs-id1278664\" type=\"1\">\n<li>Determine whether the major axis lies on the <em>x<\/em>&#8211; or <em>y<\/em>-axis.\n<ol id=\"fs-id1350073\" type=\"a\">\n<li>If the given coordinates of the vertices and foci have the form[latex]\\,\\left(\u00b1a,0\\right)\\,[\/latex]and[latex]\\,\\left(\u00b1c,0\\right)\\,[\/latex]respectively, then the major axis is the <em>x<\/em>-axis. Use the standard form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1.[\/latex]<\/li>\n<li>If the given coordinates of the vertices and foci have the form[latex]\\,\\left(0,\u00b1a\\right)\\,[\/latex]and[latex]\\,\\left(\u00b1c,0\\right),[\/latex]respectively, then the major axis is the <em>y<\/em>-axis. Use the standard form[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1.[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Use the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2},\\,[\/latex]along with the given coordinates of the vertices and foci, to solve for[latex]\\,{b}^{2}.[\/latex]<\/li>\n<li>Substitute the values for[latex]\\,{a}^{2}\\,[\/latex]and[latex]\\,{b}^{2}\\,[\/latex]into the standard form of the equation determined in Step 1.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_01_01\" class=\"textbox examples\">\n<div id=\"fs-id1326575\">\n<div id=\"fs-id1326578\">\n<h3>Writing the Equation of an Ellipse Centered at the Origin in Standard Form<\/h3>\n<p id=\"fs-id1326583\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(\u00b18,0\\right)\\,[\/latex]and foci[latex]\\,\\left(\u00b15,0\\right)?\\,[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1370498\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1370498\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1370498\">The foci are on the <em>x<\/em>-axis, so the major axis is the <em>x<\/em>-axis. Thus, the equation will have the form<\/p>\n<div id=\"fs-id1344890\" class=\"unnumbered aligncenter\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/div>\n<p id=\"fs-id1350382\">The vertices are[latex]\\,\\left(\u00b18,0\\right),[\/latex]so[latex]\\,a=8\\,[\/latex]and[latex]\\,{a}^{2}=64.[\/latex]<\/p>\n<p id=\"fs-id1339910\">The foci are[latex]\\,\\left(\u00b15,0\\right),[\/latex]so[latex]\\,c=5\\,[\/latex]and[latex]\\,{c}^{2}=25.[\/latex]<\/p>\n<p id=\"fs-id1355048\">We know that the vertices and foci are related by the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,{b}^{2},[\/latex] we have:<\/p>\n<div id=\"fs-id1362221\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}{c}^{2}={a}^{2}-{b}^{2}\\hfill & \\hfill \\\\ 25=64-{b}^{2}\\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Substitute for }{c}^{2}\\text{ and }{a}^{2}.\\hfill \\\\ {b}^{2}=39\\hfill & \\text{Solve for }{b}^{2}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1369043\">Now we need only substitute[latex]\\,{a}^{2}=64\\,[\/latex]and[latex]\\,{b}^{2}=39\\,[\/latex]into the standard form of the equation. The equation of the ellipse is[latex]\\,\\frac{{x}^{2}}{64}+\\frac{{y}^{2}}{39}=1.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1304230\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_01_01\">\n<div id=\"fs-id1340710\">\n<p id=\"fs-id1340711\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(0,\u00b14\\right)\\,[\/latex]and foci[latex]\\,\\left(0,\u00b1\\sqrt{15}\\right)?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1376849\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1376849\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1376849\">[latex]{x}^{2}+\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1347532\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1347538\"><strong>Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?<\/strong><\/p>\n<p id=\"fs-id1363857\"><em>Yes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form[latex]\\,\\left(\u00b1a,0\\right)\\,[\/latex]or[latex]\\,\\left(0,\\,\u00b1a\\right).\\,[\/latex]Similarly, the coordinates of the foci will always have the form[latex]\\,\\left(\u00b1c,0\\right)\\,[\/latex]or[latex]\\,\\left(0,\\,\u00b1c\\right).\\,[\/latex]Knowing this, we can use[latex]\\,a\\,[\/latex]and[latex]\\,c\\,[\/latex]from the given points, along with the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2},[\/latex]to find[latex]\\,{b}^{2}.[\/latex]<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1316402\" class=\"bc-section section\">\n<h4>Writing Equations of Ellipses Not Centered at the Origin<\/h4>\n<p id=\"fs-id1316030\">Like the graphs of other equations, the graph of an <span class=\"no-emphasis\">ellipse<\/span> can be translated. If an ellipse is translated[latex]\\,h\\,[\/latex]units horizontally and[latex]\\,k\\,[\/latex]units vertically, the center of the ellipse will be[latex]\\,\\left(h,k\\right).\\,[\/latex]This <span class=\"no-emphasis\">translation<\/span> results in the standard form of the equation we saw previously, with[latex]\\,x\\,[\/latex]replaced by[latex]\\,\\left(x-h\\right)\\,[\/latex]and <em>y<\/em> replaced by[latex]\\,\\left(y-k\\right).[\/latex]<\/p>\n<div id=\"fs-id1360258\" class=\"textbox key-takeaways\">\n<h3>Standard Forms of the Equation of an Ellipse with Center (<em>h<\/em>, <em>k<\/em>)<\/h3>\n<p id=\"fs-id1336434\">The standard form of the equation of an ellipse with center[latex]\\,\\left(h,\\text{ }k\\right)\\,[\/latex]and <span class=\"no-emphasis\">major axis<\/span> parallel to the <em>x<\/em>-axis is<\/p>\n<div id=\"Equation_10_01_03\">[latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/div>\n<p id=\"fs-id1355127\">where<\/p>\n<ul id=\"fs-id1355130\">\n<li>[latex]a>b[\/latex]<\/li>\n<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(h\u00b1a,k\\right)[\/latex]<\/li>\n<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(h,k\u00b1b\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(h\u00b1c,k\\right),[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_006\">(Figure)<\/a><strong>a<\/strong><\/li>\n<\/ul>\n<p id=\"fs-id1358861\">The standard form of the equation of an ellipse with center[latex]\\,\\left(h,k\\right)\\,[\/latex]and major axis parallel to the <em>y<\/em>-axis is<\/p>\n<div id=\"Equation_10_01_04\">[latex]\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1[\/latex]<\/div>\n<p id=\"fs-id1354473\">where<\/p>\n<ul id=\"fs-id1347416\">\n<li>[latex]a>b[\/latex]<\/li>\n<li>the length of the major axis is[latex]\\,2a[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(h,k\u00b1a\\right)[\/latex]<\/li>\n<li>the length of the minor axis is[latex]\\,2b[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(h\u00b1b,k\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(h,k\u00b1c\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_10_01_006\">(Figure)<\/a><strong>b<\/strong><\/li>\n<\/ul>\n<p id=\"fs-id1307508\">Just as with ellipses centered at the origin, ellipses that are centered at a point[latex]\\,\\left(h,k\\right)\\,[\/latex]have vertices, co-vertices, and foci that are related by the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]We can use this relationship along with the midpoint and distance formulas to find the equation of the ellipse in standard form when the vertices and foci are given.<\/p>\n<div id=\"Figure_10_01_006\" class=\"wp-caption aligncenter\">\n<div class=\"wp-caption-text\"><\/div>\n<div style=\"width: 945px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151241\/CNX_Precalc_Figure_10_01_006-1.jpg\" alt=\"\" width=\"935\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7. <\/strong>(a) Horizontal ellipse with center[latex]\\,\\left(h,k\\right)\\,[\/latex](b) Vertical ellipse with center[latex]\\,\\left(h,k\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1328854\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1328861\"><strong>Given the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.<\/strong><\/p>\n<ol id=\"fs-id1328866\" type=\"1\">\n<li>Determine whether the major axis is parallel to the <em>x<\/em>&#8211; or <em>y<\/em>-axis.\n<ol id=\"fs-id1332632\" type=\"a\">\n<li>If the <em>y<\/em>-coordinates of the given vertices and foci are the same, then the major axis is parallel to the <em>x<\/em>-axis. Use the standard form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1.[\/latex]<\/li>\n<li>If the <em>x<\/em>-coordinates of the given vertices and foci are the same, then the major axis is parallel to the <em>y<\/em>-axis. Use the standard form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1.[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Identify the center of the ellipse[latex]\\,\\left(h,k\\right)\\,[\/latex]using the midpoint formula and the given coordinates for the vertices.<\/li>\n<li>Find[latex]\\,{a}^{2}\\,[\/latex]by solving for the length of the major axis,[latex]\\,2a,[\/latex] which is the distance between the given vertices.<\/li>\n<li>Find[latex]\\,{c}^{2}\\,[\/latex]using[latex]\\,h\\,[\/latex]and[latex]\\,k,[\/latex] found in Step 2, along with the given coordinates for the foci.<\/li>\n<li>Solve for[latex]\\,{b}^{2}\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/li>\n<li>Substitute the values for[latex]\\,h,k,{a}^{2},[\/latex] and[latex]\\,{b}^{2}\\,[\/latex]into the standard form of the equation determined in Step 1.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_01_02\" class=\"textbox examples\">\n<div id=\"fs-id1328619\">\n<div id=\"fs-id1328621\">\n<h3>Writing the Equation of an Ellipse Centered at a Point Other Than the Origin<\/h3>\n<p id=\"fs-id1328627\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(-2,-8\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{2}\\right)[\/latex]<\/p>\n<p id=\"fs-id1343972\">and foci[latex]\\,\\left(-2,-7\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{1}\\right)?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1342912\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1342912\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1342912\">The <em>x<\/em>-coordinates of the vertices and foci are the same, so the major axis is parallel to the <em>y<\/em>-axis. Thus, the equation of the ellipse will have the form<\/p>\n<div id=\"fs-id1355444\" class=\"unnumbered aligncenter\">[latex]\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1[\/latex]<\/div>\n<p id=\"fs-id1370586\">First, we identify the center,[latex]\\,\\left(h,k\\right).\\,[\/latex]The center is halfway between the vertices,[latex]\\,\\left(-2,-8\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{2}\\right).\\,[\/latex]Applying the midpoint formula, we have:<\/p>\n<div id=\"fs-id1372947\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\left(h,k\\right)=\\left(\\frac{-2+\\left(-2\\right)}{2},\\frac{-8+2}{2}\\right)\\hfill \\\\ \\text{ }=\\left(-2,-3\\right)\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1336766\">Next, we find[latex]\\,{a}^{2}.\\,[\/latex]The length of the major axis,[latex]\\,2a,[\/latex] is bounded by the vertices. We solve for[latex]\\,a\\,[\/latex]by finding the distance between the <em>y<\/em>-coordinates of the vertices.<\/p>\n<div id=\"fs-id1359290\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2a=2-\\left(-8\\right)\\\\ 2a=10\\\\ a=5\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1367878\">So[latex]\\,{a}^{2}=25.[\/latex]<\/p>\n<p id=\"fs-id1359360\">Now we find[latex]\\,{c}^{2}.\\,[\/latex]The foci are given by[latex]\\,\\left(h,k\u00b1c\\right).\\,[\/latex]So,[latex]\\,\\left(h,k-c\\right)=\\left(-2,-7\\right)\\,[\/latex]and[latex]\\,\\left(h,k+c\\right)=\\left(-2,\\text{1}\\right).\\,[\/latex]We substitute[latex]\\,k=-3\\,[\/latex]using either of these points to solve for[latex]\\,c.[\/latex]<\/p>\n<div id=\"fs-id1329404\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\,\\,\\,\\,k+c=1\\\\ -3+c=1\\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,c=4\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1368995\">So[latex]\\,{c}^{2}=16.[\/latex]<\/p>\n<p id=\"fs-id1336896\">Next, we solve for[latex]\\,{b}^{2}\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/p>\n<div id=\"fs-id1346065\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{c}^{2}={a}^{2}-{b}^{2}\\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,16=25-{b}^{2}\\\\ {b}^{2}=9\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1385046\">Finally, we substitute the values found for[latex]\\,h,k,{a}^{2},[\/latex] and[latex]\\,{b}^{2}\\,[\/latex]into the standard form equation for an ellipse:<\/p>\n<div id=\"fs-id2289909\" class=\"unnumbered aligncenter\">[latex]\\,\\frac{{\\left(x+2\\right)}^{2}}{9}+\\frac{{\\left(y+3\\right)}^{2}}{25}=1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1346186\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_01_02\">\n<div id=\"fs-id1362367\">\n<p id=\"fs-id1362368\">What is the standard form equation of the ellipse that has vertices[latex]\\,\\left(-3,3\\right)\\,[\/latex]and[latex]\\,\\left(5,3\\right)\\,[\/latex]and foci[latex]\\,\\left(1-2\\sqrt{3},3\\right)\\,[\/latex]and[latex]\\,\\left(1+2\\sqrt{3},3\\right)?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1385633\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1385633\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1385633\">[latex]\\frac{{\\left(x-1\\right)}^{2}}{16}+\\frac{{\\left(y-3\\right)}^{2}}{4}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1291369\" class=\"bc-section section\">\n<h3>Graphing Ellipses Centered at the Origin<\/h3>\n<p id=\"fs-id1291374\">Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\text{ }a>b\\,[\/latex]for horizontal ellipses and[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,\\text{ }a>b\\,[\/latex]for vertical ellipses.<\/p>\n<div id=\"fs-id1333613\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1343715\"><strong>Given the standard form of an equation for an ellipse centered at[latex]\\,\\left(0,0\\right),[\/latex] sketch the graph.<\/strong><\/p>\n<ol id=\"fs-id1385565\" type=\"1\">\n<li>Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.\n<ol id=\"fs-id1333256\" type=\"a\">\n<li>If the equation is in the form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,a>b,\\,[\/latex]then\n<ul id=\"fs-id1334467\">\n<li>the major axis is the <em>x<\/em>-axis<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(\u00b1a,0\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(0,\u00b1b\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(\u00b1c,0\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>If the equation is in the form[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,[\/latex]where[latex]\\,a>b,\\,[\/latex]then\n<ul id=\"fs-id1333547\">\n<li>the major axis is the <em>y<\/em>-axis<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(0,\u00b1a\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(\u00b1b,0\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(0,\u00b1c\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<li>Solve for[latex]\\,c\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/li>\n<li>Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_01_03\" class=\"textbox examples\">\n<div id=\"fs-id1354398\">\n<div id=\"fs-id1346301\">\n<h3>Graphing an Ellipse Centered at the Origin<\/h3>\n<p id=\"fs-id1346306\">Graph the ellipse given by the equation,[latex]\\,\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{25}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1362387\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1362387\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1362387\">First, we determine the position of the major axis. Because[latex]\\,25>9,[\/latex]the major axis is on the <em>y<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,[\/latex]where[latex]\\,{b}^{2}=9\\,[\/latex]and[latex]\\,{a}^{2}=25.\\,[\/latex]It follows that:<\/p>\n<ul id=\"fs-id1333413\">\n<li>the center of the ellipse is[latex]\\,\\left(0,0\\right)[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(0,\u00b1a\\right)=\\left(0,\u00b1\\sqrt{25}\\right)=\\left(0,\u00b15\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(\u00b1b,0\\right)=\\left(\u00b1\\sqrt{9},0\\right)=\\left(\u00b13,0\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(0,\u00b1c\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}\\,[\/latex]Solving for[latex]\\,c,[\/latex] we have:<\/li>\n<\/ul>\n<div id=\"fs-id1328030\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{25-9}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{16}\\hfill \\\\ \\,\\,\\,=\u00b14\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1375192\">Therefore, the coordinates of the foci are[latex]\\,\\left(0,\u00b14\\right).[\/latex]<\/p>\n<p id=\"fs-id1370885\">Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_007\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_01_007\" class=\"medium\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151248\/CNX_Precalc_Figure_10_01_007.jpg\" alt=\"\" width=\"731\" height=\"521\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1366513\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_01_03\">\n<div id=\"fs-id1341581\">\n<p id=\"fs-id1341582\">Graph the ellipse given by the equation[latex]\\,\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{4}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1355523\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1355523\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1355523\">center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(\u00b16,0\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(0,\u00b12\\right);\\,[\/latex]foci:[latex]\\,\\left(\u00b14\\sqrt{2},0\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1354910\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151254\/CNX_Precalc_Figure_10_01_008.jpg\" alt=\"\" \/><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_01_04\" class=\"textbox examples\">\n<div id=\"fs-id1354926\">\n<div id=\"fs-id1354928\">\n<h3>Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form<\/h3>\n<p id=\"fs-id1354933\">Graph the ellipse given by the equation[latex]\\,4{x}^{2}+25{y}^{2}=100.\\,[\/latex]Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1343403\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1343403\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1343403\">First, use algebra to rewrite the equation in standard form.<\/p>\n<div id=\"fs-id1372134\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l} 4{x}^{2}+25{y}^{2}=100\\hfill \\\\ \\text{ }\\frac{4{x}^{2}}{100}+\\frac{25{y}^{2}}{100}=\\frac{100}{100}\\hfill \\\\ \\text{ }\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1373251\">Next, we determine the position of the major axis. Because[latex]\\,25>4,\\,[\/latex]the major axis is on the <em>x<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,{a}^{2}=25\\,[\/latex]and[latex]\\,{b}^{2}=4.\\,[\/latex]It follows that:<\/p>\n<ul id=\"fs-id1331280\">\n<li>the center of the ellipse is[latex]\\,\\left(0,0\\right)[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(\u00b1a,0\\right)=\\left(\u00b1\\sqrt{25},0\\right)=\\left(\u00b15,0\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(0,\u00b1b\\right)=\\left(0,\u00b1\\sqrt{4}\\right)=\\left(0,\u00b12\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(\u00b1c,0\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,\\,[\/latex]we have:<\/li>\n<\/ul>\n<div id=\"fs-id1355541\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{25-4}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{21}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1377751\">Therefore the coordinates of the foci are[latex]\\,\\left(\u00b1\\sqrt{21},0\\right).[\/latex]<\/p>\n<p id=\"fs-id1298485\">Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.<\/p>\n<div id=\"Figure_10_01_009\" class=\"medium\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151256\/CNX_Precalc_Figure_10_01_009.jpg\" alt=\"\" width=\"731\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1298505\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_01_04\">\n<div id=\"fs-id1334387\">\n<p id=\"fs-id1334388\">Graph the ellipse given by the equation[latex]\\,49{x}^{2}+16{y}^{2}=784.\\,[\/latex]Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1229750\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1229750\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1229750\">Standard form:[latex]\\,\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{49}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(0,\u00b17\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(\u00b14,0\\right);\\,[\/latex]foci:[latex]\\,\\left(0,\u00b1\\sqrt{33}\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1316504\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151307\/CNX_Precalc_Figure_10_01_010.jpg\" alt=\"\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1316517\" class=\"bc-section section\">\n<h3>Graphing Ellipses Not Centered at the Origin<\/h3>\n<p>When an <span class=\"no-emphasis\">ellipse<\/span> is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,[latex]\\,\\left(h,k\\right),[\/latex]we use the standard forms[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\text{ }a>b\\,[\/latex]for horizontal ellipses and[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\text{ }a>b\\,[\/latex]for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.<\/p>\n<div id=\"fs-id1298882\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1298889\"><strong>Given the standard form of an equation for an ellipse centered at[latex]\\,\\left(h,k\\right),[\/latex] sketch the graph.<\/strong><\/p>\n<ol id=\"fs-id1358186\" type=\"1\">\n<li>Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.\n<ol id=\"fs-id1358193\" type=\"a\">\n<li>If the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,a>b,\\,[\/latex]then\n<ul id=\"fs-id1367701\">\n<li>the center is[latex]\\,\\left(h,k\\right)[\/latex]<\/li>\n<li>the major axis is parallel to the <em>x<\/em>-axis<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(h\u00b1a,k\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(h,k\u00b1b\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(h\u00b1c,k\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>If the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\,[\/latex]where[latex]\\,a>b,\\,[\/latex]then\n<ul id=\"fs-id1388657\">\n<li>the center is[latex]\\,\\left(h,k\\right)[\/latex]<\/li>\n<li>the major axis is parallel to the <em>y<\/em>-axis<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(h,k\u00b1a\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(h\u00b1b,k\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(h,k\u00b1c\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<li>Solve for[latex]\\,c\\,[\/latex]using the equation[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.[\/latex]<\/li>\n<li>Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_01_05\" class=\"textbox examples\">\n<div id=\"fs-id1358880\">\n<div id=\"fs-id1358883\">\n<h3>Graphing an Ellipse Centered at (<em>h<\/em>, <em>k<\/em>)<\/h3>\n<p id=\"fs-id1361566\">Graph the ellipse given by the equation,[latex]\\,\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y-5\\right)}^{2}}{9}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1351738\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1351738\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1351738\">First, we determine the position of the major axis. Because[latex]\\,9>4,[\/latex] the major axis is parallel to the <em>y<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\,[\/latex]where[latex]\\,{b}^{2}=4\\,[\/latex]and[latex]\\,{a}^{2}=9.\\,[\/latex]It follows that:<\/p>\n<ul id=\"fs-id1290987\">\n<li>the center of the ellipse is[latex]\\,\\left(h,k\\right)=\\left(-2,\\text{5}\\right)[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(h,k\u00b1a\\right)=\\left(-2,5\u00b1\\sqrt{9}\\right)=\\left(-2,5\u00b13\\right),[\/latex] or[latex]\\,\\left(-2,\\text{2}\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{8}\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(h\u00b1b,k\\right)=\\left(-2\u00b1\\sqrt{4},5\\right)=\\left(-2\u00b12,5\\right),[\/latex] or[latex]\\,\\left(-4,5\\right)\\,[\/latex]and[latex]\\,\\left(0,\\text{5}\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(h,k\u00b1c\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,[\/latex]we have:<\/li>\n<\/ul>\n<div id=\"fs-id1368094\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\end{array}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{9-4}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{5}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1194789\">Therefore, the coordinates of the foci are[latex]\\,\\left(-2,\\text{5}-\\sqrt{5}\\right)\\,[\/latex]and[latex]\\,\\left(-2,\\text{5+}\\sqrt{5}\\right).[\/latex]<\/p>\n<p id=\"fs-id1368273\">Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.<\/p>\n<div id=\"Figure_10_01_011\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151309\/CNX_Precalc_Figure_10_01_011.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1316791\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_01_05\">\n<div id=\"fs-id1316801\">\n<p id=\"fs-id1316802\">Graph the ellipse given by the equation[latex]\\,\\frac{{\\left(x-4\\right)}^{2}}{36}+\\frac{{\\left(y-2\\right)}^{2}}{20}=1.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1361229\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1361229\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1361229\">Center:[latex]\\,\\left(4,2\\right);\\,[\/latex]vertices:[latex]\\,\\left(-2,2\\right)\\,[\/latex]and[latex]\\,\\left(10,2\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(4,2-2\\sqrt{5}\\right)\\,[\/latex]and[latex]\\,\\left(4,2+2\\sqrt{5}\\right);\\,[\/latex]foci:[latex]\\,\\left(0,2\\right)\\,[\/latex]and[latex]\\,\\left(8,2\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1365677\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151341\/CNX_Precalc_Figure_10_01_012.jpg\" alt=\"\" \/><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1365689\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1365696\"><strong>Given the general form of an equation for an ellipse centered at (<em>h<\/em>, <em>k<\/em>), express the equation in standard form.<\/strong><\/p>\n<ol id=\"fs-id1366214\" type=\"1\">\n<li>Recognize that an ellipse described by an equation in the form[latex]\\,a{x}^{2}+b{y}^{2}+cx+dy+e=0\\,[\/latex]is in general form.<\/li>\n<li>Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.<\/li>\n<li>Factor out the coefficients of the[latex]\\,{x}^{2}\\,[\/latex]and[latex]\\,{y}^{2}\\,[\/latex]terms in preparation for completing the square.<\/li>\n<li>Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant,[latex]\\,{m}_{1}{\\left(x-h\\right)}^{2}+{m}_{2}{\\left(y-k\\right)}^{2}={m}_{3},[\/latex] where[latex]\\,{m}_{1},{m}_{2},[\/latex] and[latex]\\,{m}_{3}\\,[\/latex]are constants.<\/li>\n<li>Divide both sides of the equation by the constant term to express the equation in standard form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_01_06\" class=\"textbox examples\">\n<div id=\"fs-id1353840\">\n<div id=\"fs-id1353842\">\n<h3>Graphing an Ellipse Centered at (<em>h<\/em>, <em>k<\/em>) by First Writing It in Standard Form<\/h3>\n<p id=\"fs-id1340159\">Graph the ellipse given by the equation[latex]\\,4{x}^{2}+9{y}^{2}-40x+36y+100=0.\\,[\/latex]Identify and label the center, vertices, co-vertices, and foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1366619\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1366619\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1366619\">We must begin by rewriting the equation in standard form.<\/p>\n<div id=\"fs-id1366623\" class=\"unnumbered aligncenter\">[latex]4{x}^{2}+9{y}^{2}-40x+36y+100=0[\/latex]<\/div>\n<p id=\"fs-id1367404\">Group terms that contain the same variable, and move the constant to the opposite side of the equation.<\/p>\n<div id=\"fs-id1367407\" class=\"unnumbered aligncenter\">[latex]\\left(4{x}^{2}-40x\\right)+\\left(9{y}^{2}+36y\\right)=-100[\/latex]<\/div>\n<p id=\"fs-id1350008\">Factor out the coefficients of the squared terms.<\/p>\n<div id=\"fs-id1350011\" class=\"unnumbered aligncenter\">[latex]4\\left({x}^{2}-10x\\right)+9\\left({y}^{2}+4y\\right)=-100[\/latex]<\/div>\n<p id=\"fs-id1313081\">Complete the square twice. Remember to balance the equation by adding the same constants to each side.<\/p>\n<div id=\"fs-id1313084\" class=\"unnumbered aligncenter\">[latex]4\\left({x}^{2}-10x+25\\right)+9\\left({y}^{2}+4y+4\\right)=-100+100+36[\/latex]<\/div>\n<p id=\"fs-id1360172\">Rewrite as perfect squares.<\/p>\n<div id=\"fs-id1360175\" class=\"unnumbered aligncenter\">[latex]4{\\left(x-5\\right)}^{2}+9{\\left(y+2\\right)}^{2}=36[\/latex]<\/div>\n<p id=\"fs-id1360100\">Divide both sides by the constant term to place the equation in standard form.<\/p>\n<div id=\"fs-id1360104\" class=\"unnumbered aligncenter\">[latex]\\frac{{\\left(x-5\\right)}^{2}}{9}+\\frac{{\\left(y+2\\right)}^{2}}{4}=1[\/latex]<\/div>\n<p id=\"fs-id1306079\">Now that the equation is in standard form, we can determine the position of the major axis. Because[latex]\\,9>4,\\,[\/latex] the major axis is parallel to the <em>x<\/em>-axis. Therefore, the equation is in the form[latex]\\,\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,{a}^{2}=9\\,[\/latex]and[latex]\\,{b}^{2}=4.\\,[\/latex]It follows that:<\/p>\n<ul id=\"fs-id1336814\">\n<li>the center of the ellipse is[latex]\\,\\left(h,k\\right)=\\left(5,-2\\right)[\/latex]<\/li>\n<li>the coordinates of the vertices are[latex]\\,\\left(h\u00b1a,k\\right)=\\left(5\u00b1\\sqrt{9},-2\\right)=\\left(5\u00b13,-2\\right),\\,[\/latex]or[latex]\\,\\left(2,-2\\right)\\,[\/latex]and[latex]\\,\\left(8,-2\\right)[\/latex]<\/li>\n<li>the coordinates of the co-vertices are[latex]\\,\\left(h,k\u00b1b\\right)=\\left(\\text{5},-2\u00b1\\sqrt{4}\\right)=\\left(\\text{5},-2\u00b12\\right),\\,[\/latex]or[latex]\\,\\left(5,-4\\right)\\,[\/latex]and[latex]\\,\\left(5,\\text{0}\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are[latex]\\,\\left(h\u00b1c,k\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,\\,[\/latex]we have:<\/li>\n<\/ul>\n<div id=\"fs-id1389238\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}c=\u00b1\\sqrt{{a}^{2}-{b}^{2}}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{9-4}\\hfill \\\\ \\,\\,\\,=\u00b1\\sqrt{5}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1355910\">Therefore, the coordinates of the foci are[latex]\\,\\left(\\text{5}-\\sqrt{5},-2\\right)\\,[\/latex]and[latex]\\,\\left(\\text{5+}\\sqrt{5},-2\\right).[\/latex]<\/p>\n<p>Next we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_018\">(Figure)<\/a>.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151347\/CNX_Precalc_Figure_10_01_018.jpg\" alt=\"\" width=\"487\" height=\"365\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1396393\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_01_06\">\n<div id=\"fs-id1396402\">\n<p id=\"fs-id1396403\">Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.<\/p>\n<div id=\"fs-id1320958\" class=\"unnumbered aligncenter\">[latex]4{x}^{2}+{y}^{2}-24x+2y+21=0[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1388899\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1388899\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1388899\">[latex]\\,\\frac{{\\left(x-3\\right)}^{2}}{4}+\\frac{{\\left(y+1\\right)}^{2}}{16}=1;\\,[\/latex]center:[latex]\\,\\left(3,-1\\right);\\,[\/latex]vertices:[latex]\\,\\left(3,-\\text{5}\\right)\\,[\/latex]and[latex]\\,\\left(3,\\text{3}\\right);\\,[\/latex]co-vertices:[latex]\\,\\left(1,-1\\right)\\,[\/latex]and[latex]\\,\\left(5,-1\\right);\\,[\/latex]foci:[latex]\\,\\left(3,-\\text{1}-2\\sqrt{3}\\right)\\,[\/latex]and[latex]\\,\\left(3,-\\text{1+}2\\sqrt{3}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1340629\" class=\"bc-section section\">\n<h3>Solving Applied Problems Involving Ellipses<\/h3>\n<p id=\"fs-id1340634\">Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See <a class=\"autogenerated-content\" href=\"#Figure_10_01_013\">(Figure)<\/a>. In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci\u2014about 43 feet apart\u2014can hear each other whisper.<\/p>\n<div id=\"Figure_10_01_013\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151357\/CNX_Precalc_Figure_10_01_013.jpg\" alt=\"\" width=\"487\" height=\"338\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12.<\/strong>Sound waves are reflected between foci in an elliptical room, called a whispering chamber.<\/p>\n<\/div>\n<\/div>\n<div id=\"Example_10_01_07\" class=\"textbox examples\">\n<div id=\"fs-id1377690\">\n<div id=\"fs-id1377692\">\n<h3>Locating the Foci of a Whispering Chamber<\/h3>\n<p id=\"fs-id1377697\">The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in <a class=\"autogenerated-content\" href=\"#Figure_10_01_015\">(Figure)<\/a>.<\/p>\n<ol id=\"fs-id1377705\" type=\"a\">\n<li>What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point[latex]\\,\\left(0,0\\right).[\/latex]<\/li>\n<li>If two senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.<\/li>\n<\/ol>\n<div id=\"Figure_10_01_015\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151401\/CNX_Precalc_Figure_10_01_015.jpg\" alt=\"\" width=\"487\" height=\"298\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1351971\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1351971\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1351971\" type=\"a\">\n<li>We are assuming a horizontal ellipse with center[latex]\\,\\left(0,0\\right),[\/latex] so we need to find an equation of the form[latex]\\,\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\,[\/latex]where[latex]\\,a>b.\\,[\/latex]We know that the length of the major axis,[latex]\\,2a,\\,[\/latex]is longer than the length of the minor axis,[latex]\\,2b.\\,[\/latex]So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis.\n<ul id=\"fs-id1361057\">\n<li>Solving for[latex]\\,a,[\/latex] we have[latex]\\,2a=96,[\/latex] so[latex]\\,a=48,[\/latex] and[latex]\\,{a}^{2}=2304.[\/latex]<\/li>\n<li>Solving for[latex]\\,b,[\/latex] we have[latex]\\,2b=46,[\/latex] so[latex]\\,b=23,[\/latex] and[latex]\\,{b}^{2}=529.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1396675\">Therefore, the equation of the ellipse is[latex]\\,\\frac{{x}^{2}}{2304}+\\frac{{y}^{2}}{529}=1.[\/latex]<\/p>\n<\/li>\n<li>To find the distance between the senators, we must find the distance between the foci,[latex]\\,\\left(\u00b1c,0\\right),\\,[\/latex]where[latex]\\,{c}^{2}={a}^{2}-{b}^{2}.\\,[\/latex]Solving for[latex]\\,c,[\/latex]we have:\n<div id=\"fs-id1363850\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}{c}^{2}={a}^{2}-{b}^{2}\\hfill & \\hfill \\\\ {c}^{2}=2304-529\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Substitute using the values found in part (a)}.\\hfill \\\\ \\,\\,\\,c=\u00b1\\sqrt{2304-529}\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Take the square root of both sides}.\\hfill \\\\ \\,\\,\\,c=\u00b1\\sqrt{1775} \\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Subtract}.\\hfill \\\\ \\,\\,\\,c\\approx \u00b142\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Round to the nearest foot}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1372191\">The points[latex]\\,\\left(\u00b142,0\\right)\\,[\/latex]represent the foci. Thus, the distance between the senators is[latex]\\,2\\left(42\\right)=84\\,[\/latex]feet.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1377180\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_01_07\">\n<div id=\"fs-id1377190\">\n<p id=\"fs-id1377191\">Suppose a whispering chamber is 480 feet long and 320 feet wide.<\/p>\n<ol id=\"fs-id1377194\" type=\"a\">\n<li>What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point[latex]\\,\\left(0,0\\right).[\/latex]<\/li>\n<li>If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1403959\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1403959\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1403959\" type=\"a\">\n<li>[latex]\\frac{{x}^{2}}{57,600}+\\frac{{y}^{2}}{25,600}=1[\/latex]<\/li>\n<li>The people are standing 358 feet apart.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1315937\" class=\"precalculus media\">\n<p id=\"fs-id1315944\">Access these online resources for additional instruction and practice with ellipses.<\/p>\n<ul id=\"fs-id1315947\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/conicellipse\">Conic Sections: The Ellipse<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/grphellorigin\">Graph an Ellipse with Center at the Origin<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/grphellnot\">Graph an Ellipse with Center Not at the Origin<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1351649\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1351656\" summary=\"..\">\n<tbody>\n<tr>\n<td>Horizontal ellipse, center at origin<\/td>\n<td>[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Vertical ellipse, center at origin<\/td>\n<td>[latex]\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Horizontal ellipse, center [latex]\\,\\left(h,k\\right)[\/latex]<\/td>\n<td>[latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Vertical ellipse, center[latex]\\,\\left(h,k\\right)[\/latex]<\/td>\n<td>[latex]\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1368433\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1368439\">\n<li>An ellipse is the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).<\/li>\n<li>When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See <a class=\"autogenerated-content\" href=\"#Example_10_01_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_10_01_02\">(Figure)<\/a>.<\/li>\n<li>When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See <a class=\"autogenerated-content\" href=\"#Example_10_01_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_10_01_04\">(Figure)<\/a>.<\/li>\n<li>When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See <a class=\"autogenerated-content\" href=\"#Example_10_01_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_10_01_06\">(Figure)<\/a>.<\/li>\n<li>Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See <a class=\"autogenerated-content\" href=\"#Example_10_01_07\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1343126\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1343129\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1343134\">\n<div id=\"fs-id1343136\">\n<p id=\"fs-id1343137\">Define an ellipse in terms of its foci.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1374799\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1374799\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1374799\">An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1374803\">\n<div id=\"fs-id1374804\">\n<p id=\"fs-id1374805\">Where must the foci of an ellipse lie?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1374809\">\n<div id=\"fs-id1374810\">\n<p id=\"fs-id1374811\">What special case of the ellipse do we have when the major and minor axis are of the same length?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1374815\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1374815\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1374815\">This special case would be a circle.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1374818\">\n<div id=\"fs-id1374819\">\n<p id=\"fs-id1374820\">For the special case mentioned above, what would be true about the foci of that ellipse?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1374824\">\n<div id=\"fs-id1374825\">\n<p id=\"fs-id1374826\">What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the <em>y<\/em>-axis?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1374835\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1374835\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1374835\">It is symmetric about the <em>x<\/em>-axis, <em>y<\/em>-axis, and the origin.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1351258\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1351264\">For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.<\/p>\n<div id=\"fs-id1351268\">\n<div id=\"fs-id1351269\">\n<p id=\"fs-id1351270\">[latex]2{x}^{2}+y=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1351300\">\n<div id=\"fs-id1351301\">\n<p id=\"fs-id1351302\">[latex]4{x}^{2}+9{y}^{2}=36[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1194849\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1194849\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1194849\">yes;[latex]\\,\\frac{{x}^{2}}{{3}^{2}}+\\frac{{y}^{2}}{{2}^{2}}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1339824\">\n<div id=\"fs-id1339826\">\n<p id=\"fs-id1339827\">[latex]4{x}^{2}-{y}^{2}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1339864\">\n<div id=\"fs-id1339865\">\n<p id=\"fs-id1339866\">[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1377100\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1377100\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1377100\">yes;[latex]\\frac{{x}^{2}}{{\\left(\\frac{1}{2}\\right)}^{2}}+\\frac{{y}^{2}}{{\\left(\\frac{1}{3}\\right)}^{2}}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1316571\">\n<div id=\"fs-id1316572\">\n<p id=\"fs-id1316573\">[latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1343327\">For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.<\/p>\n<div id=\"fs-id1350501\">\n<div id=\"fs-id1350502\">\n<p id=\"fs-id1350503\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{49}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1320315\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1320315\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1320315\">[latex]\\frac{{x}^{2}}{{2}^{2}}+\\frac{{y}^{2}}{{7}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(0,7\\right)\\,[\/latex]and[latex]\\,\\left(0,-7\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(2,0\\right)\\,[\/latex]and[latex]\\,\\left(-2,0\\right).\\,[\/latex]Foci at[latex]\\,\\left(0,3\\sqrt{5}\\right),\\left(0,-3\\sqrt{5}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1346718\">\n<div id=\"fs-id1346719\">\n<p id=\"fs-id1346720\">[latex]\\frac{{x}^{2}}{100}+\\frac{{y}^{2}}{64}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1359968\">\n<div id=\"fs-id1359969\">\n<p id=\"fs-id1359970\">[latex]{x}^{2}+9{y}^{2}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1385319\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1385319\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1385319\">[latex]\\frac{{x}^{2}}{{\\left(1\\right)}^{2}}+\\frac{{y}^{2}}{{\\left(\\frac{1}{3}\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(1,0\\right)\\,[\/latex]and[latex]\\,\\left(-1,0\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(0,\\frac{1}{3}\\right),\\left(0,-\\frac{1}{3}\\right).\\,[\/latex]Foci at[latex]\\,\\left(\\frac{2\\sqrt{2}}{3},0\\right),\\left(-\\frac{2\\sqrt{2}}{3},0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1369209\">\n<div id=\"fs-id1369210\">\n<p id=\"fs-id1369212\">[latex]4{x}^{2}+16{y}^{2}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1374887\">\n<div id=\"fs-id1374888\">\n<p id=\"fs-id1374889\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{49}+\\frac{{\\left(y-4\\right)}^{2}}{25}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1363641\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1363641\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1363641\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{{7}^{2}}+\\frac{{\\left(y-4\\right)}^{2}}{{5}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(9,4\\right),\\left(-5,4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(2,9\\right),\\left(2,-1\\right).\\,[\/latex]Foci at[latex]\\,\\left(2+2\\sqrt{6},4\\right),\\left(2-2\\sqrt{6},4\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1376228\">\n<div id=\"fs-id1376229\">\n<p id=\"fs-id1376230\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{81}+\\frac{{\\left(y+1\\right)}^{2}}{16}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1349361\">\n<div id=\"fs-id1349362\">\n<p id=\"fs-id1349363\">[latex]\\frac{{\\left(x+5\\right)}^{2}}{4}+\\frac{{\\left(y-7\\right)}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1359220\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1359220\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1359220\">[latex]\\frac{{\\left(x+5\\right)}^{2}}{{2}^{2}}+\\frac{{\\left(y-7\\right)}^{2}}{{3}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(-5,10\\right),\\left(-5,4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(-3,7\\right),\\left(-7,7\\right).\\,[\/latex]Foci at[latex]\\,\\left(-5,7+\\sqrt{5}\\right),\\left(-5,7-\\sqrt{5}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1389365\">\n<div id=\"fs-id1389366\">\n<p id=\"fs-id1389367\">[latex]\\frac{{\\left(x-7\\right)}^{2}}{49}+\\frac{{\\left(y-7\\right)}^{2}}{49}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1340052\">\n<div id=\"fs-id1340053\">\n<p id=\"fs-id1340054\">[latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1365819\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1365819\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1365819\">[latex]\\frac{{\\left(x-1\\right)}^{2}}{{3}^{2}}+\\frac{{\\left(y-4\\right)}^{2}}{{2}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(4,4\\right),\\left(-2,4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(1,6\\right),\\left(1,2\\right).\\,[\/latex]Foci at[latex]\\,\\left(1+\\sqrt{5},4\\right),\\left(1-\\sqrt{5},4\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1326905\">\n<div id=\"fs-id1326906\">\n<p id=\"fs-id1326907\">[latex]9{x}^{2}-54x+9{y}^{2}-54y+81=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1371116\">\n<div id=\"fs-id1371117\">\n<p id=\"fs-id1371118\">[latex]4{x}^{2}-24x+36{y}^{2}-360y+864=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1342974\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1342974\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1342974\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{{\\left(3\\sqrt{2}\\right)}^{2}}+\\frac{{\\left(y-5\\right)}^{2}}{{\\left(\\sqrt{2}\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(3+3\\sqrt{2},5\\right),\\left(3-3\\sqrt{2},5\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(3,5+\\sqrt{2}\\right),\\left(3,5-\\sqrt{2}\\right).\\,[\/latex]Foci at[latex]\\,\\left(7,5\\right),\\left(-1,5\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id693861\">\n<div id=\"fs-id693862\">\n<p id=\"fs-id693863\">[latex]4{x}^{2}+24x+16{y}^{2}-128y+228=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1364961\">\n<div id=\"fs-id1364962\">\n<p id=\"fs-id1364963\">[latex]4{x}^{2}+40x+25{y}^{2}-100y+100=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1361188\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1361188\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1361188\">[latex]\\frac{{\\left(x+5\\right)}^{2}}{{\\left(5\\right)}^{2}}+\\frac{{\\left(y-2\\right)}^{2}}{{\\left(2\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(0,2\\right),\\left(-10,2\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(-5,4\\right),\\left(-5,0\\right).\\,[\/latex]Foci at[latex]\\,\\left(-5+\\sqrt{21},2\\right),\\left(-5-\\sqrt{21},2\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1332177\">\n<div id=\"fs-id1332178\">\n<p id=\"fs-id1332179\">[latex]{x}^{2}+2x+100{y}^{2}-1000y+2401=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1313180\">\n<div id=\"fs-id1313181\">\n<p id=\"fs-id1313182\">[latex]4{x}^{2}+24x+25{y}^{2}+200y+336=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1332424\">Show Solution<\/span><\/p>\n<div id=\"q1332424\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{{\\left(x+3\\right)}^{2}}{{\\left(5\\right)}^{2}}+\\frac{{\\left(y+4\\right)}^{2}}{{\\left(2\\right)}^{2}}=1;\\,[\/latex]Endpoints of major axis[latex]\\,\\left(2,-4\\right),\\left(-8,-4\\right).\\,[\/latex]Endpoints of minor axis[latex]\\,\\left(-3,-2\\right),\\left(-3,-6\\right).\\,[\/latex]Foci at[latex]\\,\\left(-3+\\sqrt{21},-4\\right),\\left(-3-\\sqrt{21},-4\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1327267\">\n<div id=\"fs-id1327268\">\n<p id=\"fs-id1327270\">[latex]9{x}^{2}+72x+16{y}^{2}+16y+4=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1332231\">For the following exercises, find the foci for the given ellipses.<\/p>\n<div id=\"fs-id1332234\">\n<div id=\"fs-id1332235\">\n<p id=\"fs-id1332236\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{25}+\\frac{{\\left(y+1\\right)}^{2}}{36}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1332043\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1332043\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1332043\">Foci[latex]\\,\\left(-3,-1+\\sqrt{11}\\right),\\left(-3,-1-\\sqrt{11}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1313640\">\n<div id=\"fs-id1313641\">\n<p id=\"fs-id1313642\">[latex]\\frac{{\\left(x+1\\right)}^{2}}{100}+\\frac{{\\left(y-2\\right)}^{2}}{4}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1396602\">\n<div id=\"fs-id1396603\">\n<p id=\"fs-id1396604\">[latex]{x}^{2}+{y}^{2}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1396641\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1396641\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1396641\">Focus[latex]\\,\\left(0,0\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1328074\">\n<div id=\"fs-id1328075\">\n<p id=\"fs-id1328076\">[latex]{x}^{2}+4{y}^{2}+4x+8y=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1328126\">\n<div id=\"fs-id1328127\">\n<p id=\"fs-id1328128\">[latex]10{x}^{2}+{y}^{2}+200x=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1346558\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1346558\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1346558\">Foci[latex]\\,\\left(-10,30\\right),\\left(-10,-30\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1377000\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1377005\">For the following exercises, graph the given ellipses, noting center, vertices, and foci.<\/p>\n<div id=\"fs-id1377008\">\n<div id=\"fs-id1377009\">\n<p id=\"fs-id1377010\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{36}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1331349\">\n<div id=\"fs-id1331350\">\n<p id=\"fs-id1331351\">[latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1368187\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1368187\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1368187\">Center[latex]\\,\\left(0,0\\right),\\,[\/latex]Vertices[latex]\\,\\left(4,0\\right),\\left(-4,0\\right),\\left(0,3\\right),\\left(0,-3\\right),\\,[\/latex]Foci[latex]\\,\\left(\\sqrt{7},0\\right),\\left(-\\sqrt{7},0\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1360065\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151404\/CNX_Precalc_Figure_10_01_202.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1360074\">\n<div id=\"fs-id1360075\">\n<p id=\"fs-id1360076\">[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1404169\">\n<div id=\"fs-id1404170\">\n<p id=\"fs-id1404171\">[latex]81{x}^{2}+49{y}^{2}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1404211\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1404211\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1404211\">Center[latex]\\,\\left(0,0\\right),\\,[\/latex]Vertices[latex]\\,\\left(\\frac{1}{9},0\\right),\\left(-\\frac{1}{9},0\\right),\\left(0,\\frac{1}{7}\\right),\\left(0,-\\frac{1}{7}\\right),\\,[\/latex]Foci[latex]\\,\\left(0,\\frac{4\\sqrt{2}}{63}\\right),\\left(0,-\\frac{4\\sqrt{2}}{63}\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1329173\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151419\/CNX_Precalc_Figure_10_01_204.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1329183\">\n<div id=\"fs-id1329184\">\n<p id=\"fs-id1329185\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{64}+\\frac{{\\left(y-4\\right)}^{2}}{16}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1326325\">\n<div id=\"fs-id1326326\">\n<p id=\"fs-id1326327\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{9}+\\frac{{\\left(y-3\\right)}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1388768\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1388768\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1388768\">Center[latex]\\,\\left(-3,3\\right),\\,[\/latex]Vertices[latex]\\,\\left(0,3\\right),\\left(-6,3\\right),\\left(-3,0\\right),\\left(-3,6\\right),\\,[\/latex]Focus[latex]\\,\\left(-3,3\\right)\\,[\/latex]<\/p>\n<p id=\"fs-id1327102\">Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.<\/p>\n<p><span id=\"fs-id1327108\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151434\/CNX_Precalc_Figure_10_01_206.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1354710\">\n<div id=\"fs-id1354711\">\n<p id=\"fs-id1354712\">[latex]\\frac{{x}^{2}}{2}+\\frac{{\\left(y+1\\right)}^{2}}{5}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1332813\">\n<div id=\"fs-id1332814\">\n<p id=\"fs-id1332815\">[latex]4{x}^{2}-8x+16{y}^{2}-32y-44=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1332872\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1332872\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1332872\">Center[latex]\\,\\left(1,1\\right),\\,[\/latex]Vertices[latex]\\,\\left(5,1\\right),\\left(-3,1\\right),\\left(1,3\\right),\\left(1,-1\\right),\\,[\/latex]Foci[latex]\\,\\left(1,1+4\\sqrt{3}\\right),\\left(1,1-4\\sqrt{3}\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1363950\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151437\/CNX_Precalc_Figure_10_01_208.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1363960\">\n<div id=\"fs-id1363961\">\n<p id=\"fs-id1363962\">[latex]{x}^{2}-8x+25{y}^{2}-100y+91=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1326410\">\n<div id=\"fs-id1326411\">\n<p id=\"fs-id1326412\">[latex]{x}^{2}+8x+4{y}^{2}-40y+112=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1326468\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1326468\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1326468\">Center[latex]\\,\\left(-4,5\\right),\\,[\/latex]Vertices[latex]\\,\\left(-2,5\\right),\\left(-6,4\\right),\\left(-4,6\\right),\\left(-4,4\\right),\\,[\/latex]Foci[latex]\\,\\left(-4+\\sqrt{3},5\\right),\\left(-4-\\sqrt{3},5\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1344024\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151443\/CNX_Precalc_Figure_10_01_210.jpg\" alt=\"\" \/><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1344033\">\n<div id=\"fs-id1344034\">\n<p id=\"fs-id1344036\">[latex]64{x}^{2}+128x+9{y}^{2}-72y-368=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1333303\">\n<div id=\"fs-id1333304\">\n<p id=\"fs-id1333305\">[latex]16{x}^{2}+64x+4{y}^{2}-8y+4=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1194702\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1194702\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1194702\">Center[latex]\\,\\left(-2,1\\right),\\,[\/latex]Vertices[latex]\\,\\left(0,1\\right),\\left(-4,1\\right),\\left(-2,5\\right),\\left(-2,-3\\right),\\,[\/latex]Foci[latex]\\,\\left(-2,1+2\\sqrt{3}\\right),\\left(-2,1-2\\sqrt{3}\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1396328\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151449\/CNX_Precalc_Figure_10_01_212.jpg\" alt=\"\" \/><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1396338\">\n<div id=\"fs-id1396339\">\n<p id=\"fs-id1396340\">[latex]100{x}^{2}+1000x+{y}^{2}-10y+2425=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1363717\">\n<div id=\"fs-id1363718\">\n<p id=\"fs-id1363719\">[latex]4{x}^{2}+16x+4{y}^{2}+16y+16=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1326715\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1326715\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1326715\">Center[latex]\\,\\left(-2,-2\\right),\\,[\/latex]Vertices[latex]\\,\\left(0,-2\\right),\\left(-4,-2\\right),\\left(-2,0\\right),\\left(-2,-4\\right),\\,[\/latex]Focus[latex]\\,\\left(-2,-2\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1327616\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151451\/CNX_Precalc_Figure_10_01_214.jpg\" alt=\"\" \/><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1327626\">For the following exercises, use the given information about the graph of each ellipse to determine its equation.<\/p>\n<div id=\"fs-id1327630\">\n<div id=\"fs-id1327631\">\n<p id=\"fs-id1327632\">Center at the origin, symmetric with respect to the <em>x<\/em>&#8211; and <em>y<\/em>-axes, focus at[latex]\\,\\left(4,0\\right),[\/latex] and point on graph[latex]\\,\\left(0,3\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1304004\">\n<div id=\"fs-id1304005\">\n<p id=\"fs-id1304006\">Center at the origin, symmetric with respect to the <em>x<\/em>&#8211; and <em>y<\/em>-axes, focus at[latex]\\,\\left(0,-2\\right),[\/latex] and point on graph[latex]\\,\\left(5,0\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1368635\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1368635\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1368635\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{29}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1368697\">\n<div id=\"fs-id1368698\">\n<p id=\"fs-id1368699\">Center at the origin, symmetric with respect to the <em>x<\/em>&#8211; and <em>y<\/em>-axes, focus at[latex]\\,\\left(3,0\\right),[\/latex] and major axis is twice as long as minor axis.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1320506\">\n<div id=\"fs-id1320507\">\n<p id=\"fs-id1320508\">Center[latex]\\,\\left(4,2\\right)[\/latex]; vertex[latex]\\,\\left(9,2\\right)[\/latex]; one focus:[latex]\\,\\left(4+2\\sqrt{6},2\\right)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1364314\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1364314\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1364314\">[latex]\\frac{{\\left(x-4\\right)}^{2}}{25}+\\frac{{\\left(y-2\\right)}^{2}}{1}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1303537\">\n<div id=\"fs-id1303538\">\n<p id=\"fs-id1303539\">Center[latex]\\,\\left(3,5\\right)[\/latex]; vertex[latex]\\,\\left(3,11\\right)[\/latex]; one focus:[latex]\\,\\left(3,\\text{ 5+4}\\sqrt{\\text{2}}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1346675\">\n<div id=\"fs-id1346676\">\n<p id=\"fs-id1346678\">Center[latex]\\,\\left(-3,4\\right)[\/latex]; vertex[latex]\\,\\left(1,4\\right)[\/latex]; one focus:[latex]\\,\\left(-3+2\\sqrt{3},4\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1396512\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1396512\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1396512\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{16}+\\frac{{\\left(y-4\\right)}^{2}}{4}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1343632\">For the following exercises, given the graph of the ellipse, determine its equation.<\/p>\n<div id=\"fs-id1343635\">\n<div id=\"fs-id1343636\"><span id=\"fs-id1319617\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151500\/CNX_Precalc_Figure_10_01_215.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1319627\">\n<div id=\"fs-id1319628\"><span id=\"fs-id1319633\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151513\/CNX_Precalc_Figure_10_01_216.jpg\" alt=\"\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1319644\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1319644\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1319644\">[latex]\\frac{{x}^{2}}{81}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1349436\">\n<div id=\"fs-id1349437\"><span id=\"fs-id1349442\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151515\/CNX_Precalc_Figure_10_01_217.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1349453\">\n<div id=\"fs-id1349454\"><span id=\"fs-id1349459\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151517\/CNX_Precalc_Figure_10_01_218.jpg\" alt=\"\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1349470\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1349470\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1349470\">[latex]\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y-2\\right)}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1402281\">\n<div id=\"fs-id1402282\"><span id=\"fs-id1402287\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19151523\/CNX_Precalc_Figure_10_01_219.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1402299\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1402304\">For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula[latex]\\,\\text{Area}=a\\cdot b\\cdot \\pi .[\/latex]<\/p>\n<div id=\"fs-id1345055\">\n<div id=\"fs-id1345056\">\n<p id=\"fs-id1345058\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{9}+\\frac{{\\left(y-3\\right)}^{2}}{16}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1365738\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1365738\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1365738\">[latex]\\text{Area = 12\u03c0}\\,\\text{square}\\,\\text{units}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1365758\">\n<div id=\"fs-id1365759\">\n<p id=\"fs-id1365760\">[latex]\\frac{{\\left(x+6\\right)}^{2}}{16}+\\frac{{\\left(y-6\\right)}^{2}}{36}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1359643\">\n<div id=\"fs-id1359644\">\n<p id=\"fs-id1359646\">[latex]\\frac{{\\left(x+1\\right)}^{2}}{4}+\\frac{{\\left(y-2\\right)}^{2}}{5}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1343831\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1343831\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1343831\">[latex]\\text{Area = 2}\\sqrt{\\text{5}}\\text{\u03c0}\\,\\text{square}\\,\\text{units}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1343860\">\n<div id=\"fs-id1343861\">\n<p id=\"fs-id1319526\">[latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1319582\">\n<div id=\"fs-id1319583\">\n<p id=\"fs-id1319584\">[latex]9{x}^{2}-54x+9{y}^{2}-54y+81=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1404247\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1404247\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1404247\">[latex]\\text{Area = 9\u03c0}\\,\\text{square}\\,\\text{units}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1404269\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div>\n<div id=\"fs-id1404275\">\n<p id=\"fs-id1404276\">Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1404280\">\n<div id=\"fs-id1404282\">\n<p id=\"fs-id1404283\">Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of[latex]\\,h,[\/latex] the height.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1404304\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1404304\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1404304\">[latex]\\frac{{x}^{2}}{4{h}^{2}}+\\frac{{y}^{2}}{\\frac{1}{4}{h}^{2}}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1377342\">\n<div id=\"fs-id1377343\">\n<p id=\"fs-id1377344\">An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of the arch at a distance of 4 feet from the center.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1377350\">\n<div id=\"fs-id1377351\">\n<p id=\"fs-id1377352\">An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1352044\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1352044\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1352044\">[latex]\\frac{{x}^{2}}{400}+\\frac{{y}^{2}}{144}=1[\/latex]. Distance = 17.32 feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1352104\">\n<div id=\"fs-id1352106\">\n<p id=\"fs-id1352107\">A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1352112\">\n<div id=\"fs-id1352113\">\n<p id=\"fs-id1352114\">A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1352121\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1352121\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1352121\">Approximately 51.96 feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1352125\">\n<div id=\"fs-id1352126\">\n<p id=\"fs-id1352127\">A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and height at the center of the gallery?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1352137\">\n<dt>center of an ellipse<\/dt>\n<dd id=\"fs-id1345665\">the midpoint of both the major and minor axes<\/dd>\n<\/dl>\n<dl id=\"fs-id1345668\">\n<dt>conic section<\/dt>\n<dd id=\"fs-id1345672\">any shape resulting from the intersection of a right circular cone with a plane<\/dd>\n<\/dl>\n<dl id=\"fs-id1345675\">\n<dt>ellipse<\/dt>\n<dd id=\"fs-id1345680\">the set of all points[latex]\\,\\left(x,y\\right)\\,[\/latex]in a plane such that the sum of their distances from two fixed points is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1345712\">\n<dt>foci<\/dt>\n<dd id=\"fs-id1345716\">plural of focus<\/dd>\n<\/dl>\n<dl id=\"fs-id1345719\">\n<dt>focus (of an ellipse)<\/dt>\n<dd id=\"fs-id1345723\">one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point[latex]\\,\\left(x,y\\right)\\,[\/latex]on the ellipse is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1345756\">\n<dt>major axis<\/dt>\n<dd id=\"fs-id1350559\">the longer of the two axes of an ellipse<\/dd>\n<\/dl>\n<dl id=\"fs-id1350562\">\n<dt>minor axis<\/dt>\n<dd id=\"fs-id1350566\">the shorter of the two axes of an ellipse<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3229\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Algebra and Trigonometry. <strong>Authored by<\/strong>: Jay Abramson, et. al. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\">http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-3229-1\">Architect of the Capitol. http:\/\/www.aoc.gov. Accessed April 15, 2014. <a href=\"#return-footnote-3229-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":53384,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Algebra and Trigonometry\",\"author\":\"Jay Abramson, et. al\",\"organization\":\"OpenStax CNX\",\"url\":\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3229","chapter","type-chapter","status-publish","hentry"],"part":3198,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3229","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/users\/53384"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3229\/revisions"}],"predecessor-version":[{"id":3751,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3229\/revisions\/3751"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/parts\/3198"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3229\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/media?parent=3229"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapter-type?post=3229"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/contributor?post=3229"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/license?post=3229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}