{"id":1855,"date":"2021-07-28T20:25:10","date_gmt":"2021-07-28T20:25:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1855"},"modified":"2022-03-21T23:24:20","modified_gmt":"2022-03-21T23:24:20","slug":"alternative-ways-of-identifying-conic-sections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/alternative-ways-of-identifying-conic-sections\/","title":{"raw":"Alternative Ways of Identifying Conic Sections","rendered":"Alternative Ways of Identifying Conic Sections"},"content":{"raw":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Recognize a parabola, ellipse, or hyperbola from its eccentricity value<\/li>\r\n \t<li>Write the polar equation of a conic section with eccentricity [latex]e[\/latex]<\/li>\r\n \t<li>Identify when a general equation of degree two is a parabola, ellipse, or hyperbola<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Eccentricity and Directrix<\/h2>\r\n<p id=\"fs-id1167793372216\">An alternative way to describe a conic section involves the directrices, the foci, and a new property called eccentricity. We will see that the value of the eccentricity of a conic section can uniquely define that conic.<\/p>\r\n\r\n<div id=\"fs-id1167793372221\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793372227\">The <strong>eccentricity<\/strong> <em data-effect=\"italics\">e<\/em> of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. This value is constant for any conic section, and can define the conic section as well:<\/p>\r\n\r\n<ol id=\"fs-id1167793372240\" type=\"1\">\r\n \t<li>If [latex]e=1[\/latex], the conic is a parabola.<\/li>\r\n \t<li>If [latex]e&lt;1[\/latex], it is an ellipse.<\/li>\r\n \t<li>If [latex]e&gt;1[\/latex], it is a hyperbola.<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1167793372290\">The eccentricity of a circle is zero. The directrix of a conic section is the line that, together with the point known as the focus, serves to define a conic section. Hyperbolas and noncircular ellipses have two foci and two associated directrices. Parabolas have one focus and one directrix.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793372297\">The three conic sections with their directrices appear in the following figure.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_11_05_016\"><figcaption><\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"965\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225405\/CNX_Calc_Figure_11_05_016.jpg\" alt=\"This figure has three figures. In the first is an ellipse, with center at the origin, foci at (c, 0) and (\u2212c, 0), half of its vertical height being b, half of its horizontal length being a, and directrix x = \u00b1a2\/c. The second figure is a parabola with vertex at the origin, focus (a, 0), and directrix x = \u2212a. The third figure is a hyperbola with center at the origin, foci at (c, 0) and (\u2212c, 0), vertices at (a, 0) and (\u2212a, 0), and directices at x = \u00b1a2\/c.\" width=\"965\" height=\"445\" data-media-type=\"image\/jpeg\" \/> Figure 16. The three conic sections with their foci and directrices.[\/caption]<\/figure>\r\n<p id=\"fs-id1167793372322\">Recall from the definition of a parabola that the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix. Therefore, by definition, the eccentricity of a parabola must be 1. The equations of the directrices of a horizontal ellipse are [latex]x=\\pm \\frac{{a}^{2}}{c}[\/latex]. The right vertex of the ellipse is located at [latex]\\left(a,0\\right)[\/latex] and the right focus is [latex]\\left(c,0\\right)[\/latex]. Therefore the distance from the vertex to the focus is [latex]a-c[\/latex] and the distance from the vertex to the right directrix is [latex]\\frac{{a}^{2}}{c}-c[\/latex]. This gives the eccentricity as<\/p>\r\n\r\n<div id=\"fs-id1167793387343\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]e=\\frac{a-c}{\\frac{{a}^{2}}{c}-a}=\\frac{c\\left(a-c\\right)}{{a}^{2}-ac}=\\frac{c\\left(a-c\\right)}{a\\left(a-c\\right)}=\\frac{c}{a}[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1167793290965\">Since [latex]c&lt;a[\/latex], this step proves that the eccentricity of an ellipse is less than 1. The directrices of a horizontal hyperbola are also located at [latex]x=\\pm \\frac{{a}^{2}}{c}[\/latex], and a similar calculation shows that the eccentricity of a hyperbola is also [latex]e=\\frac{c}{a}[\/latex]. However in this case we have [latex]c&gt;a[\/latex], so the eccentricity of a hyperbola is greater than 1.<\/p>\r\n\r\n<div id=\"fs-id1167793291030\" data-type=\"example\">\r\n<div id=\"fs-id1167793291032\" data-type=\"exercise\">\r\n<div id=\"fs-id1167793291034\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining Eccentricity of a Conic Section<\/h3>\r\n<div id=\"fs-id1167793291034\" data-type=\"problem\">\r\n<p id=\"fs-id1167793291039\">Determine the eccentricity of the ellipse described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167793291042\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{{\\left(x - 3\\right)}^{2}}{16}+\\frac{{\\left(y+2\\right)}^{2}}{25}=1[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558890\"]\r\n<div id=\"fs-id1167793368119\" data-type=\"solution\">\r\n<p id=\"fs-id1167793368121\">From the equation we see that [latex]a=5[\/latex] and [latex]b=4[\/latex]. The value of <em data-effect=\"italics\">c<\/em> can be calculated using the equation [latex]{a}^{2}={b}^{2}+{c}^{2}[\/latex] for an ellipse. Substituting the values of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> and solving for <em data-effect=\"italics\">c<\/em> gives [latex]c=3[\/latex]. Therefore the eccentricity of the ellipse is [latex]e=\\frac{c}{a}=\\frac{3}{5}=0.6[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Determining Eccentricity of a Conic Section.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AF_0mpKk4-Y?controls=0&amp;start=1477&amp;end=1546&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.5ConicSections1477to1546_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"7.5 Conic Sections\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1167794138954\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1167794138957\" data-type=\"exercise\">\r\n<div id=\"fs-id1167794138959\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1167794138959\" data-type=\"problem\">\r\n<p id=\"fs-id1167794138961\">Determine the eccentricity of the hyperbola described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167794138965\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{{\\left(y - 3\\right)}^{2}}{49}-\\frac{{\\left(x+2\\right)}^{2}}{25}=1[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558879\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558879\"]\r\n<div id=\"fs-id1167794139057\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167794069860\">First find the values of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, then determine <em data-effect=\"italics\">c<\/em> using the equation [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558889\"]\r\n<div id=\"fs-id1167794139025\" data-type=\"solution\">\r\n<p id=\"fs-id1167794139028\" style=\"text-align: center;\">[latex]e=\\frac{c}{a}=\\frac{\\sqrt{74}}{7}\\approx 1.229[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]146740[\/ohm_question]\r\n\r\n<\/div>\r\n<section id=\"fs-id1167794069905\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Polar Equations of Conic Sections<\/h2>\r\n<p id=\"fs-id1167794069911\">Sometimes it is useful to write or identify the equation of a conic section in polar form. To do this, we need the concept of the focal parameter. The <span data-type=\"term\">focal parameter<\/span> of a conic section <em data-effect=\"italics\">p<\/em> is defined as the distance from a focus to the nearest directrix. The following table gives the focal parameters for the different types of conics, where <em data-effect=\"italics\">a<\/em> is the length of the semi-major axis (i.e., half the length of the major axis), <em data-effect=\"italics\">c<\/em> is the distance from the origin to the focus, and <em data-effect=\"italics\">e<\/em> is the eccentricity. In the case of a parabola, <em data-effect=\"italics\">a<\/em> represents the distance from the vertex to the focus.<\/p>\r\n\r\n<table id=\"fs-id1167794069945\" summary=\"This table has three columns and four rows. The first row is a header row and reads from left to right Conic, e, and p. After the header, the first column reads Ellipse, Parabola, and Hyperbola. The second column reads 0 &lt; e &lt; 1, e = 1, and e &gt; 1. The third column reads (a2 \u2013 c2)\/c = a(1 \u2013 e2)\/c, 2a, and (c2 \u2013 a2)\/c = a(e2 \u2013 1)\/e.\"><caption><span data-type=\"title\">Eccentricities and Focal Parameters of the Conic Sections<\/span><\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-valign=\"top\" data-align=\"left\">Conic<\/th>\r\n<th data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">e<\/em><\/th>\r\n<th data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">p<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Ellipse<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">[latex]0&lt;e&lt;1[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">[latex]\\frac{{a}^{2}-{c}^{2}}{c}=\\frac{a\\left(1-{e}^{2}\\right)}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Parabola<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">[latex]e=1[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">[latex]2a[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Hyperbola<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">[latex]e&gt;1[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">[latex]\\frac{{c}^{2}-{a}^{2}}{c}=\\frac{a\\left({e}^{2}-1\\right)}{e}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167793377603\">Using the definitions of the focal parameter and eccentricity of the conic section, we can derive an equation for any conic section in polar coordinates. In particular, we assume that one of the foci of a given conic section lies at the pole. Then using the definition of the various conic sections in terms of distances, it is possible to prove the following theorem.<\/p>\r\n\r\n<div id=\"fs-id1167793377609\" class=\"theorem\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">theorem: Polar Equation of Conic Sections<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793377616\">The polar equation of a conic section with focal parameter <em data-effect=\"italics\">p<\/em> is given by<\/p>\r\n\r\n<div id=\"fs-id1167793377623\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\dfrac{ep}{1\\pm e\\cos\\theta }\\text{ or }r=\\dfrac{ep}{1\\pm e\\sin\\theta }[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793377684\">In the equation on the left, the major axis of the conic section is horizontal, and in the equation on the right, the major axis is vertical. To work with a conic section written in polar form, first make the constant term in the denominator equal to 1. This can be done by dividing both the numerator and the denominator of the fraction by the constant that appears in front of the plus or minus in the denominator. Then the coefficient of the sine or cosine in the denominator is the eccentricity. This value identifies the conic. If cosine appears in the denominator, then the conic is horizontal. If sine appears, then the conic is vertical. If both appear then the axes are rotated. The center of the conic is not necessarily at the origin. The center is at the origin only if the conic is a circle (i.e., [latex]e=0[\/latex]).<\/p>\r\n\r\n<div id=\"fs-id1167793377708\" data-type=\"example\">\r\n<div id=\"fs-id1167793377710\" data-type=\"exercise\">\r\n<div id=\"fs-id1167793377712\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Conic Section in Polar Coordinates<\/h3>\r\n<div id=\"fs-id1167793377712\" data-type=\"problem\">\r\n<p id=\"fs-id1167793377718\">Identify and create a graph of the conic section described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167793377721\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\dfrac{3}{1+2\\cos\\theta }[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558869\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558869\"]\r\n<div id=\"fs-id1167793361326\" data-type=\"solution\">\r\n<p id=\"fs-id1167793361328\">The constant term in the denominator is 1, so the eccentricity of the conic is 2. This is a hyperbola. The focal parameter <em data-effect=\"italics\">p<\/em> can be calculated by using the equation [latex]ep=3[\/latex]. Since [latex]e=2[\/latex], this gives [latex]p=\\frac{3}{2}[\/latex]. The cosine function appears in the denominator, so the hyperbola is horizontal. Pick a few values for [latex]\\theta [\/latex] and create a table of values. Then we can graph the hyperbola (Figure 17).<\/p>\r\n\r\n<table id=\"fs-id1167793361392\" class=\"unnumbered\" summary=\"This table has two columns and nine rows. The first row is a header row and reads from left to right \u03b8 and 4. After the header, the first column reads 0, \u03c0\/4, \u03c0\/2, 3\u03c0\/4, \u03c0, 5\u03c0\/4, 3\u03c0\/2, and 7\u03c0\/4. The second column reads 1, 3 divided by the quantity (1 + the square root of 2), which is approximately equal to 1.2426, 3, 3 divided by the quantity (1 \u2013 the square root of 2), which is approximately equal to \u22127.2426, \u22123, 3 divided by the quantity (1 \u2013 the square root of 2), which is approximately equal to \u22127.2426, 3, and 3 divided by the quantity (1 + the square root of 2), which is approximately equal to 1.2426.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-valign=\"top\" data-align=\"center\">[latex]\\theta [\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]r[\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]\\theta [\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]r[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\pi [\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1+\\sqrt{2}}\\approx 1.2426[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{5\\pi }{4}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1-\\sqrt{2}}\\approx -7.2426[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1-\\sqrt{2}}\\approx -7.2426[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{7\\pi }{4}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1+\\sqrt{2}}\\approx 1.2426[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<figure id=\"CNX_Calc_Figure_11_05_017\">[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225408\/CNX_Calc_Figure_11_05_017.jpg\" alt=\"Graph of a hyperbola with equation r = 3\/(1 + 2 cos\u03b8), center at (2, 0), and vertices at (1, 0) and (3, 0).\" width=\"417\" height=\"497\" data-media-type=\"image\/jpeg\" \/> Figure 17. Graph of the hyperbola described in [link].[\/caption]<\/figure>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Graphing a Conic Section in Polar Coordinates.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AF_0mpKk4-Y?controls=0&amp;start=1665&amp;end=1820&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.5ConicSections1665to1820_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"7.5 Conic Sections\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1167793380144\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1167793380148\" data-type=\"exercise\">\r\n<div id=\"fs-id1167793380150\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1167793380150\" data-type=\"problem\">\r\n<p id=\"fs-id1167793380152\">Identify and create a graph of the conic section described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167793380155\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\frac{4}{1 - 0.8\\sin\\theta }[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558849\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558849\"]\r\n<div id=\"fs-id1167794172253\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167794172261\">First find the values of <em data-effect=\"italics\">e<\/em> and <em data-effect=\"italics\">p<\/em>, and then create a table of values.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558859\"]\r\n<div id=\"fs-id1167793380185\" data-type=\"solution\">\r\n<p id=\"fs-id1167793380187\">Here [latex]e=0.8[\/latex] and [latex]p=5[\/latex]. This conic section is an ellipse.<span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"424\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225411\/CNX_Calc_Figure_11_05_018.jpg\" alt=\"Graph of an ellipse with equation r = 4\/(1 \u2013 0.8 sin\u03b8), center near (0, 11), major axis roughly 22, and minor axis roughly 12.\" width=\"424\" height=\"572\" data-media-type=\"image\/jpeg\" \/> Figure 18.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]25566[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1167794172279\" data-depth=\"1\">\r\n<h2 data-type=\"title\">General Equations of Degree Two<\/h2>\r\n<p id=\"fs-id1167794172285\">A general equation of degree two can be written in the form<\/p>\r\n\r\n<div id=\"fs-id1167794172288\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1167794172345\">The graph of an equation of this form is a conic section. If [latex]B\\ne 0[\/latex] then the coordinate axes are rotated. To identify the conic section, we use the <strong>discriminant<\/strong> of the conic section [latex]4AC-{B}^{2}[\/latex]. One of the following cases must be true:<\/p>\r\n\r\n<ol id=\"fs-id1167794172380\" type=\"1\">\r\n \t<li>[latex]4AC-{B}^{2}&gt;0[\/latex]. If so, the graph is an ellipse.<\/li>\r\n \t<li>[latex]4AC-{B}^{2}=0[\/latex]. If so, the graph is a parabola.<\/li>\r\n \t<li>[latex]4AC-{B}^{2}&lt;0[\/latex]. If so, the graph is a hyperbola.<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1167793395219\">The simplest example of a second-degree equation involving a cross term is [latex]xy=1[\/latex]. This equation can be solved for <em data-effect=\"italics\">y<\/em> to obtain [latex]y=\\frac{1}{x}[\/latex]. The graph of this function is called a <em data-effect=\"italics\">rectangular hyperbola<\/em> as shown.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_11_05_019\"><figcaption><\/figcaption>[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225413\/CNX_Calc_Figure_11_05_019.jpg\" alt=\"Graph of xy = 1, which has asymptotes at the x and y axes. This hyperbola is relegated to the first and third quadrants, and the graph also has red dashed lines along y = x and y = \u2212x.\" width=\"417\" height=\"422\" data-media-type=\"image\/jpeg\" \/> Figure 19. Graph of the equation [latex]xy=1[\/latex]; The red lines indicate the rotated axes.[\/caption]<\/figure>\r\n<p id=\"fs-id1167793395296\">The asymptotes of this hyperbola are the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> coordinate axes. To determine the angle [latex]\\theta [\/latex] of rotation of the conic section, we use the formula [latex]\\cot{2\\theta}=\\frac{A-C}{B}[\/latex]. In this case [latex]A=C=0[\/latex] and [latex]B=1[\/latex], so [latex]\\cot{2\\theta}=\\frac{\\left(0 - 0\\right)}{1}=0[\/latex] and [latex]\\theta =45^\\circ[\/latex]. The method for graphing a conic section with rotated axes involves determining the coefficients of the conic in the rotated coordinate system. The new coefficients are labeled [latex]{A}^{\\prime },{B}^{\\prime },{C}^{\\prime },{D}^{\\prime },{E}^{\\prime },\\text{ and }{F}^{\\prime }[\/latex], and are given by the formulas<\/p>\r\n\r\n<div id=\"fs-id1167793382549\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill {A}^{\\prime }&amp; =\\hfill &amp; A{\\cos}^{2}\\theta +B\\cos\\theta \\sin\\theta +C{\\sin}^{2}\\theta \\hfill \\\\ \\hfill {B}^{\\prime }&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill {C}^{\\prime }&amp; =\\hfill &amp; A{\\sin}^{2}\\theta -B\\sin\\theta \\cos\\theta +C{\\cos}^{2}\\theta \\hfill \\\\ \\hfill {D}^{\\prime }&amp; =\\hfill &amp; D\\cos\\theta +E\\sin\\theta \\hfill \\\\ \\hfill {E}^{\\prime }&amp; =\\hfill &amp; -D\\sin\\theta +E\\cos\\theta \\hfill \\\\ \\hfill {F}^{\\prime }&amp; =\\hfill &amp; F.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1167794142352\">The procedure for graphing a rotated conic is the following:<\/p>\r\n\r\n<ol id=\"fs-id1167794142355\" type=\"1\">\r\n \t<li>Identify the conic section using the discriminant [latex]4AC-{B}^{2}[\/latex].<\/li>\r\n \t<li>Determine [latex]\\theta [\/latex] using the formula [latex]\\cot2\\theta =\\frac{A-C}{B}[\/latex].<\/li>\r\n \t<li>Calculate [latex]{A}^{\\prime },{B}^{\\prime },{C}^{\\prime },{D}^{\\prime },{E}^{\\prime },\\text{and}{F}^{\\prime }[\/latex].<\/li>\r\n \t<li>Rewrite the original equation using [latex]{A}^{\\prime },{B}^{\\prime },{C}^{\\prime },{D}^{\\prime },{E}^{\\prime },\\text{and}{F}^{\\prime }[\/latex].<\/li>\r\n \t<li>Draw a graph using the rotated equation.<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1167793991353\" data-type=\"example\">\r\n<div id=\"fs-id1167793991356\" data-type=\"exercise\">\r\n<div id=\"fs-id1167793991358\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying a Rotated Conic<\/h3>\r\n<div id=\"fs-id1167793991358\" data-type=\"problem\">\r\n<p id=\"fs-id1167793991363\">Identify the conic and calculate the angle of rotation of axes for the curve described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167793991367\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}-256=0[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558839\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558839\"]\r\n<div id=\"fs-id1167793991415\" data-type=\"solution\">\r\n<p id=\"fs-id1167793991417\">In this equation, [latex]A=13,B=-6\\sqrt{3},C=7,D=0,E=0[\/latex], and [latex]F=-256[\/latex]. The discriminant of this equation is [latex]4AC-{B}^{2}=4\\left(13\\right)\\left(7\\right)-{\\left(-6\\sqrt{3}\\right)}^{2}=364 - 108=256[\/latex]. Therefore this conic is an ellipse. To calculate the angle of rotation of the axes, use [latex]\\cot2\\theta =\\frac{A-C}{B}[\/latex]. This gives<\/p>\r\n\r\n<div id=\"fs-id1167794119002\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}\\hfill \\cot2\\theta &amp; =\\frac{A-C}{B}\\hfill \\\\ &amp; =\\frac{13 - 7}{-6\\sqrt{3}}\\hfill \\\\ &amp; =-\\frac{\\sqrt{3}}{3}.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1167794119077\">Therefore [latex]2 \\theta = {120}^{\\circ}[\/latex] and [latex]\\theta ={60}^{\\circ}[\/latex], which is the angle of the rotation of the axes.<\/p>\r\n<p id=\"fs-id1167794119112\">To determine the rotated coefficients, use the formulas given above:<\/p>\r\n\r\n<div id=\"fs-id1167794119115\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill {A}^{\\prime }&amp; =\\hfill &amp; A{\\cos}^{2}\\theta +B\\cos\\theta \\sin\\theta +C{\\sin}^{2}\\theta \\hfill \\\\ &amp; =\\hfill &amp; 13{\\cos}^{2}60+\\left(-6\\sqrt{3}\\right)\\cos60\\sin60+7{\\sin}^{2}60\\hfill \\\\ &amp; =\\hfill &amp; 13{\\left(\\frac{1}{2}\\right)}^{2}-6\\sqrt{3}\\left(\\frac{1}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right)+7{\\left(\\frac{\\sqrt{3}}{2}\\right)}^{2}\\hfill \\\\ &amp; =\\hfill &amp; 4,\\hfill \\\\ \\hfill {B}^{\\prime }&amp; =\\hfill &amp; 0,\\hfill \\\\ \\hfill {C}^{\\prime }&amp; =\\hfill &amp; A{\\sin}^{2}\\theta -B\\sin\\theta \\cos\\theta +C{\\cos}^{2}\\theta \\hfill \\\\ &amp; =\\hfill &amp; 13{\\sin}^{2}60+\\left(-6\\sqrt{3}\\right)\\sin60\\cos60=7{\\cos}^{2}60\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\left(\\frac{\\sqrt{3}}{2}\\right)}^{2}+6\\sqrt{3}\\left(\\frac{\\sqrt{3}}{2}\\right)\\left(\\frac{1}{2}\\right)+7{\\left(\\frac{1}{2}\\right)}^{2}\\hfill \\\\ &amp; =\\hfill &amp; 16,\\hfill \\\\ \\hfill {D}^{\\prime }&amp; =\\hfill &amp; D\\cos\\theta +E\\sin\\theta \\hfill \\\\ &amp; =\\hfill &amp; \\left(0\\right)\\cos60+\\left(0\\right)\\sin60\\hfill \\\\ &amp; =\\hfill &amp; 0,\\hfill \\\\ \\hfill {E}^{\\prime }&amp; =\\hfill &amp; -D\\sin\\theta +E\\cos\\theta \\hfill \\\\ &amp; =\\hfill &amp; -\\left(0\\right)\\sin60+\\left(0\\right)\\cos60\\hfill \\\\ &amp; =\\hfill &amp; 0,\\hfill \\\\ \\hfill {F}^{\\prime }&amp; =\\hfill &amp; F\\hfill \\\\ &amp; =\\hfill &amp; -256.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1167794136912\">The equation of the conic in the rotated coordinate system becomes<\/p>\r\n\r\n<div id=\"fs-id1167794136915\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{}\\\\ \\hfill 4{\\left({x}^{\\prime }\\right)}^{2}+16{\\left({y}^{\\prime }\\right)}^{2}&amp; =\\hfill &amp; 256\\hfill \\\\ \\hfill \\frac{{\\left({x}^{\\prime }\\right)}^{2}}{64}+\\frac{{\\left({y}^{\\prime }\\right)}^{2}}{16}&amp; =\\hfill &amp; 1.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1167793389848\">A graph of this conic section appears as follows.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_11_05_020\"><figcaption><\/figcaption>[caption id=\"\" align=\"aligncenter\" width=\"419\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225415\/CNX_Calc_Figure_11_05_020.jpg\" alt=\"Graph of an ellipse with equation 13x2 \u2013 6 times the square root of 3 times xy + 7y2 \u2013 256 = 0. The center is at the origin, and the ellipse appears to be skewed 60 degrees. There are dashed red lines along the major and minor axes.\" width=\"419\" height=\"422\" data-media-type=\"image\/jpeg\" \/> Figure 20. Graph of the ellipse described by the equation [latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}-256=0[\/latex]. The axes are rotated [latex]60^\\circ[\/latex]. The red dashed lines indicate the rotated axes.[\/caption]<\/figure>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Identifying a Rotated Conic.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AF_0mpKk4-Y?controls=0&amp;start=1939&amp;end=2314&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.5ConicSections1939to2314_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"7.5 Conic Sections\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1167793389930\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1167793389935\" data-type=\"exercise\">\r\n<div id=\"fs-id1167793389937\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1167793389937\" data-type=\"problem\">\r\n<p id=\"fs-id1167793389939\">Identify the conic and calculate the angle of rotation of axes for the curve described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167793389943\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]3{x}^{2}+5xy - 2{y}^{2}-125=0[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558819\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558819\"]\r\n<div id=\"fs-id1167793390007\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167793390014\">Follow steps 1 and 2 of the five-step method outlined above.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558829\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558829\"]\r\n<div id=\"fs-id1167793389988\" data-type=\"solution\">\r\n<p id=\"fs-id1167793389990\">The conic is a hyperbola and the angle of rotation of the axes is [latex]\\theta =22.5^{\\circ }[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]175311[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1167793390022\" class=\"key-concepts\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\"><\/div>","rendered":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Recognize a parabola, ellipse, or hyperbola from its eccentricity value<\/li>\n<li>Write the polar equation of a conic section with eccentricity [latex]e[\/latex]<\/li>\n<li>Identify when a general equation of degree two is a parabola, ellipse, or hyperbola<\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Eccentricity and Directrix<\/h2>\n<p id=\"fs-id1167793372216\">An alternative way to describe a conic section involves the directrices, the foci, and a new property called eccentricity. We will see that the value of the eccentricity of a conic section can uniquely define that conic.<\/p>\n<div id=\"fs-id1167793372221\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1167793372227\">The <strong>eccentricity<\/strong> <em data-effect=\"italics\">e<\/em> of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. This value is constant for any conic section, and can define the conic section as well:<\/p>\n<ol id=\"fs-id1167793372240\" type=\"1\">\n<li>If [latex]e=1[\/latex], the conic is a parabola.<\/li>\n<li>If [latex]e<1[\/latex], it is an ellipse.<\/li>\n<li>If [latex]e>1[\/latex], it is a hyperbola.<\/li>\n<\/ol>\n<p id=\"fs-id1167793372290\">The eccentricity of a circle is zero. The directrix of a conic section is the line that, together with the point known as the focus, serves to define a conic section. Hyperbolas and noncircular ellipses have two foci and two associated directrices. Parabolas have one focus and one directrix.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793372297\">The three conic sections with their directrices appear in the following figure.<\/p>\n<figure id=\"CNX_Calc_Figure_11_05_016\"><figcaption><\/figcaption><div style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225405\/CNX_Calc_Figure_11_05_016.jpg\" alt=\"This figure has three figures. In the first is an ellipse, with center at the origin, foci at (c, 0) and (\u2212c, 0), half of its vertical height being b, half of its horizontal length being a, and directrix x = \u00b1a2\/c. The second figure is a parabola with vertex at the origin, focus (a, 0), and directrix x = \u2212a. The third figure is a hyperbola with center at the origin, foci at (c, 0) and (\u2212c, 0), vertices at (a, 0) and (\u2212a, 0), and directices at x = \u00b1a2\/c.\" width=\"965\" height=\"445\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 16. The three conic sections with their foci and directrices.<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1167793372322\">Recall from the definition of a parabola that the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix. Therefore, by definition, the eccentricity of a parabola must be 1. The equations of the directrices of a horizontal ellipse are [latex]x=\\pm \\frac{{a}^{2}}{c}[\/latex]. The right vertex of the ellipse is located at [latex]\\left(a,0\\right)[\/latex] and the right focus is [latex]\\left(c,0\\right)[\/latex]. Therefore the distance from the vertex to the focus is [latex]a-c[\/latex] and the distance from the vertex to the right directrix is [latex]\\frac{{a}^{2}}{c}-c[\/latex]. This gives the eccentricity as<\/p>\n<div id=\"fs-id1167793387343\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]e=\\frac{a-c}{\\frac{{a}^{2}}{c}-a}=\\frac{c\\left(a-c\\right)}{{a}^{2}-ac}=\\frac{c\\left(a-c\\right)}{a\\left(a-c\\right)}=\\frac{c}{a}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167793290965\">Since [latex]c<a[\/latex], this step proves that the eccentricity of an ellipse is less than 1. The directrices of a horizontal hyperbola are also located at [latex]x=\\pm \\frac{{a}^{2}}{c}[\/latex], and a similar calculation shows that the eccentricity of a hyperbola is also [latex]e=\\frac{c}{a}[\/latex]. However in this case we have [latex]c>a[\/latex], so the eccentricity of a hyperbola is greater than 1.<\/p>\n<div id=\"fs-id1167793291030\" data-type=\"example\">\n<div id=\"fs-id1167793291032\" data-type=\"exercise\">\n<div id=\"fs-id1167793291034\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example: Determining Eccentricity of a Conic Section<\/h3>\n<div id=\"fs-id1167793291034\" data-type=\"problem\">\n<p id=\"fs-id1167793291039\">Determine the eccentricity of the ellipse described by the equation<\/p>\n<div id=\"fs-id1167793291042\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{{\\left(x - 3\\right)}^{2}}{16}+\\frac{{\\left(y+2\\right)}^{2}}{25}=1[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558890\">Show Solution<\/span><\/p>\n<div id=\"q44558890\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793368119\" data-type=\"solution\">\n<p id=\"fs-id1167793368121\">From the equation we see that [latex]a=5[\/latex] and [latex]b=4[\/latex]. The value of <em data-effect=\"italics\">c<\/em> can be calculated using the equation [latex]{a}^{2}={b}^{2}+{c}^{2}[\/latex] for an ellipse. Substituting the values of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> and solving for <em data-effect=\"italics\">c<\/em> gives [latex]c=3[\/latex]. Therefore the eccentricity of the ellipse is [latex]e=\\frac{c}{a}=\\frac{3}{5}=0.6[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Determining Eccentricity of a Conic Section.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AF_0mpKk4-Y?controls=0&amp;start=1477&amp;end=1546&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.5ConicSections1477to1546_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;7.5 Conic Sections&#8221; here (opens in new window)<\/a>.<\/p>\n<div id=\"fs-id1167794138954\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1167794138957\" data-type=\"exercise\">\n<div id=\"fs-id1167794138959\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1167794138959\" data-type=\"problem\">\n<p id=\"fs-id1167794138961\">Determine the eccentricity of the hyperbola described by the equation<\/p>\n<div id=\"fs-id1167794138965\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{{\\left(y - 3\\right)}^{2}}{49}-\\frac{{\\left(x+2\\right)}^{2}}{25}=1[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558879\">Hint<\/span><\/p>\n<div id=\"q44558879\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794139057\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167794069860\">First find the values of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, then determine <em data-effect=\"italics\">c<\/em> using the equation [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558889\">Show Solution<\/span><\/p>\n<div id=\"q44558889\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794139025\" data-type=\"solution\">\n<p id=\"fs-id1167794139028\" style=\"text-align: center;\">[latex]e=\\frac{c}{a}=\\frac{\\sqrt{74}}{7}\\approx 1.229[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146740\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146740&theme=oea&iframe_resize_id=ohm146740&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<section id=\"fs-id1167794069905\" data-depth=\"1\">\n<h2 data-type=\"title\">Polar Equations of Conic Sections<\/h2>\n<p id=\"fs-id1167794069911\">Sometimes it is useful to write or identify the equation of a conic section in polar form. To do this, we need the concept of the focal parameter. The <span data-type=\"term\">focal parameter<\/span> of a conic section <em data-effect=\"italics\">p<\/em> is defined as the distance from a focus to the nearest directrix. The following table gives the focal parameters for the different types of conics, where <em data-effect=\"italics\">a<\/em> is the length of the semi-major axis (i.e., half the length of the major axis), <em data-effect=\"italics\">c<\/em> is the distance from the origin to the focus, and <em data-effect=\"italics\">e<\/em> is the eccentricity. In the case of a parabola, <em data-effect=\"italics\">a<\/em> represents the distance from the vertex to the focus.<\/p>\n<table id=\"fs-id1167794069945\" summary=\"This table has three columns and four rows. The first row is a header row and reads from left to right Conic, e, and p. After the header, the first column reads Ellipse, Parabola, and Hyperbola. The second column reads 0 &lt; e &lt; 1, e = 1, and e &gt; 1. The third column reads (a2 \u2013 c2)\/c = a(1 \u2013 e2)\/c, 2a, and (c2 \u2013 a2)\/c = a(e2 \u2013 1)\/e.\">\n<caption><span data-type=\"title\">Eccentricities and Focal Parameters of the Conic Sections<\/span><\/caption>\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Conic<\/th>\n<th data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">e<\/em><\/th>\n<th data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">p<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Ellipse<\/td>\n<td data-valign=\"top\" data-align=\"left\">[latex]0<e<1[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"left\">[latex]\\frac{{a}^{2}-{c}^{2}}{c}=\\frac{a\\left(1-{e}^{2}\\right)}{c}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Parabola<\/td>\n<td data-valign=\"top\" data-align=\"left\">[latex]e=1[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"left\">[latex]2a[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Hyperbola<\/td>\n<td data-valign=\"top\" data-align=\"left\">[latex]e>1[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"left\">[latex]\\frac{{c}^{2}-{a}^{2}}{c}=\\frac{a\\left({e}^{2}-1\\right)}{e}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167793377603\">Using the definitions of the focal parameter and eccentricity of the conic section, we can derive an equation for any conic section in polar coordinates. In particular, we assume that one of the foci of a given conic section lies at the pole. Then using the definition of the various conic sections in terms of distances, it is possible to prove the following theorem.<\/p>\n<div id=\"fs-id1167793377609\" class=\"theorem\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">theorem: Polar Equation of Conic Sections<\/h3>\n<hr \/>\n<p id=\"fs-id1167793377616\">The polar equation of a conic section with focal parameter <em data-effect=\"italics\">p<\/em> is given by<\/p>\n<div id=\"fs-id1167793377623\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\dfrac{ep}{1\\pm e\\cos\\theta }\\text{ or }r=\\dfrac{ep}{1\\pm e\\sin\\theta }[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793377684\">In the equation on the left, the major axis of the conic section is horizontal, and in the equation on the right, the major axis is vertical. To work with a conic section written in polar form, first make the constant term in the denominator equal to 1. This can be done by dividing both the numerator and the denominator of the fraction by the constant that appears in front of the plus or minus in the denominator. Then the coefficient of the sine or cosine in the denominator is the eccentricity. This value identifies the conic. If cosine appears in the denominator, then the conic is horizontal. If sine appears, then the conic is vertical. If both appear then the axes are rotated. The center of the conic is not necessarily at the origin. The center is at the origin only if the conic is a circle (i.e., [latex]e=0[\/latex]).<\/p>\n<div id=\"fs-id1167793377708\" data-type=\"example\">\n<div id=\"fs-id1167793377710\" data-type=\"exercise\">\n<div id=\"fs-id1167793377712\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Conic Section in Polar Coordinates<\/h3>\n<div id=\"fs-id1167793377712\" data-type=\"problem\">\n<p id=\"fs-id1167793377718\">Identify and create a graph of the conic section described by the equation<\/p>\n<div id=\"fs-id1167793377721\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\dfrac{3}{1+2\\cos\\theta }[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558869\">Show Solution<\/span><\/p>\n<div id=\"q44558869\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793361326\" data-type=\"solution\">\n<p id=\"fs-id1167793361328\">The constant term in the denominator is 1, so the eccentricity of the conic is 2. This is a hyperbola. The focal parameter <em data-effect=\"italics\">p<\/em> can be calculated by using the equation [latex]ep=3[\/latex]. Since [latex]e=2[\/latex], this gives [latex]p=\\frac{3}{2}[\/latex]. The cosine function appears in the denominator, so the hyperbola is horizontal. Pick a few values for [latex]\\theta[\/latex] and create a table of values. Then we can graph the hyperbola (Figure 17).<\/p>\n<table id=\"fs-id1167793361392\" class=\"unnumbered\" summary=\"This table has two columns and nine rows. The first row is a header row and reads from left to right \u03b8 and 4. After the header, the first column reads 0, \u03c0\/4, \u03c0\/2, 3\u03c0\/4, \u03c0, 5\u03c0\/4, 3\u03c0\/2, and 7\u03c0\/4. The second column reads 1, 3 divided by the quantity (1 + the square root of 2), which is approximately equal to 1.2426, 3, 3 divided by the quantity (1 \u2013 the square root of 2), which is approximately equal to \u22127.2426, \u22123, 3 divided by the quantity (1 \u2013 the square root of 2), which is approximately equal to \u22127.2426, 3, and 3 divided by the quantity (1 + the square root of 2), which is approximately equal to 1.2426.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"center\">[latex]\\theta[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]r[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]\\theta[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]r[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\pi[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1+\\sqrt{2}}\\approx 1.2426[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{5\\pi }{4}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1-\\sqrt{2}}\\approx -7.2426[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1-\\sqrt{2}}\\approx -7.2426[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{7\\pi }{4}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3}{1+\\sqrt{2}}\\approx 1.2426[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"CNX_Calc_Figure_11_05_017\">\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225408\/CNX_Calc_Figure_11_05_017.jpg\" alt=\"Graph of a hyperbola with equation r = 3\/(1 + 2 cos\u03b8), center at (2, 0), and vertices at (1, 0) and (3, 0).\" width=\"417\" height=\"497\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 17. Graph of the hyperbola described in [link].<\/p>\n<\/div>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Graphing a Conic Section in Polar Coordinates.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AF_0mpKk4-Y?controls=0&amp;start=1665&amp;end=1820&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.5ConicSections1665to1820_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;7.5 Conic Sections&#8221; here (opens in new window)<\/a>.<\/p>\n<div id=\"fs-id1167793380144\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1167793380148\" data-type=\"exercise\">\n<div id=\"fs-id1167793380150\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1167793380150\" data-type=\"problem\">\n<p id=\"fs-id1167793380152\">Identify and create a graph of the conic section described by the equation<\/p>\n<div id=\"fs-id1167793380155\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\frac{4}{1 - 0.8\\sin\\theta }[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558849\">Hint<\/span><\/p>\n<div id=\"q44558849\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794172253\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167794172261\">First find the values of <em data-effect=\"italics\">e<\/em> and <em data-effect=\"italics\">p<\/em>, and then create a table of values.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558859\">Show Solution<\/span><\/p>\n<div id=\"q44558859\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793380185\" data-type=\"solution\">\n<p id=\"fs-id1167793380187\">Here [latex]e=0.8[\/latex] and [latex]p=5[\/latex]. This conic section is an ellipse.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div style=\"width: 434px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225411\/CNX_Calc_Figure_11_05_018.jpg\" alt=\"Graph of an ellipse with equation r = 4\/(1 \u2013 0.8 sin\u03b8), center near (0, 11), major axis roughly 22, and minor axis roughly 12.\" width=\"424\" height=\"572\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 18.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm25566\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25566&theme=oea&iframe_resize_id=ohm25566&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1167794172279\" data-depth=\"1\">\n<h2 data-type=\"title\">General Equations of Degree Two<\/h2>\n<p id=\"fs-id1167794172285\">A general equation of degree two can be written in the form<\/p>\n<div id=\"fs-id1167794172288\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167794172345\">The graph of an equation of this form is a conic section. If [latex]B\\ne 0[\/latex] then the coordinate axes are rotated. To identify the conic section, we use the <strong>discriminant<\/strong> of the conic section [latex]4AC-{B}^{2}[\/latex]. One of the following cases must be true:<\/p>\n<ol id=\"fs-id1167794172380\" type=\"1\">\n<li>[latex]4AC-{B}^{2}>0[\/latex]. If so, the graph is an ellipse.<\/li>\n<li>[latex]4AC-{B}^{2}=0[\/latex]. If so, the graph is a parabola.<\/li>\n<li>[latex]4AC-{B}^{2}<0[\/latex]. If so, the graph is a hyperbola.<\/li>\n<\/ol>\n<p id=\"fs-id1167793395219\">The simplest example of a second-degree equation involving a cross term is [latex]xy=1[\/latex]. This equation can be solved for <em data-effect=\"italics\">y<\/em> to obtain [latex]y=\\frac{1}{x}[\/latex]. The graph of this function is called a <em data-effect=\"italics\">rectangular hyperbola<\/em> as shown.<\/p>\n<figure id=\"CNX_Calc_Figure_11_05_019\"><figcaption><\/figcaption><div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225413\/CNX_Calc_Figure_11_05_019.jpg\" alt=\"Graph of xy = 1, which has asymptotes at the x and y axes. This hyperbola is relegated to the first and third quadrants, and the graph also has red dashed lines along y = x and y = \u2212x.\" width=\"417\" height=\"422\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 19. Graph of the equation [latex]xy=1[\/latex]; The red lines indicate the rotated axes.<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1167793395296\">The asymptotes of this hyperbola are the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> coordinate axes. To determine the angle [latex]\\theta[\/latex] of rotation of the conic section, we use the formula [latex]\\cot{2\\theta}=\\frac{A-C}{B}[\/latex]. In this case [latex]A=C=0[\/latex] and [latex]B=1[\/latex], so [latex]\\cot{2\\theta}=\\frac{\\left(0 - 0\\right)}{1}=0[\/latex] and [latex]\\theta =45^\\circ[\/latex]. The method for graphing a conic section with rotated axes involves determining the coefficients of the conic in the rotated coordinate system. The new coefficients are labeled [latex]{A}^{\\prime },{B}^{\\prime },{C}^{\\prime },{D}^{\\prime },{E}^{\\prime },\\text{ and }{F}^{\\prime }[\/latex], and are given by the formulas<\/p>\n<div id=\"fs-id1167793382549\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill {A}^{\\prime }& =\\hfill & A{\\cos}^{2}\\theta +B\\cos\\theta \\sin\\theta +C{\\sin}^{2}\\theta \\hfill \\\\ \\hfill {B}^{\\prime }& =\\hfill & 0\\hfill \\\\ \\hfill {C}^{\\prime }& =\\hfill & A{\\sin}^{2}\\theta -B\\sin\\theta \\cos\\theta +C{\\cos}^{2}\\theta \\hfill \\\\ \\hfill {D}^{\\prime }& =\\hfill & D\\cos\\theta +E\\sin\\theta \\hfill \\\\ \\hfill {E}^{\\prime }& =\\hfill & -D\\sin\\theta +E\\cos\\theta \\hfill \\\\ \\hfill {F}^{\\prime }& =\\hfill & F.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167794142352\">The procedure for graphing a rotated conic is the following:<\/p>\n<ol id=\"fs-id1167794142355\" type=\"1\">\n<li>Identify the conic section using the discriminant [latex]4AC-{B}^{2}[\/latex].<\/li>\n<li>Determine [latex]\\theta[\/latex] using the formula [latex]\\cot2\\theta =\\frac{A-C}{B}[\/latex].<\/li>\n<li>Calculate [latex]{A}^{\\prime },{B}^{\\prime },{C}^{\\prime },{D}^{\\prime },{E}^{\\prime },\\text{and}{F}^{\\prime }[\/latex].<\/li>\n<li>Rewrite the original equation using [latex]{A}^{\\prime },{B}^{\\prime },{C}^{\\prime },{D}^{\\prime },{E}^{\\prime },\\text{and}{F}^{\\prime }[\/latex].<\/li>\n<li>Draw a graph using the rotated equation.<\/li>\n<\/ol>\n<div id=\"fs-id1167793991353\" data-type=\"example\">\n<div id=\"fs-id1167793991356\" data-type=\"exercise\">\n<div id=\"fs-id1167793991358\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example: Identifying a Rotated Conic<\/h3>\n<div id=\"fs-id1167793991358\" data-type=\"problem\">\n<p id=\"fs-id1167793991363\">Identify the conic and calculate the angle of rotation of axes for the curve described by the equation<\/p>\n<div id=\"fs-id1167793991367\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}-256=0[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558839\">Show Solution<\/span><\/p>\n<div id=\"q44558839\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793991415\" data-type=\"solution\">\n<p id=\"fs-id1167793991417\">In this equation, [latex]A=13,B=-6\\sqrt{3},C=7,D=0,E=0[\/latex], and [latex]F=-256[\/latex]. The discriminant of this equation is [latex]4AC-{B}^{2}=4\\left(13\\right)\\left(7\\right)-{\\left(-6\\sqrt{3}\\right)}^{2}=364 - 108=256[\/latex]. Therefore this conic is an ellipse. To calculate the angle of rotation of the axes, use [latex]\\cot2\\theta =\\frac{A-C}{B}[\/latex]. This gives<\/p>\n<div id=\"fs-id1167794119002\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}\\hfill \\cot2\\theta & =\\frac{A-C}{B}\\hfill \\\\ & =\\frac{13 - 7}{-6\\sqrt{3}}\\hfill \\\\ & =-\\frac{\\sqrt{3}}{3}.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167794119077\">Therefore [latex]2 \\theta = {120}^{\\circ}[\/latex] and [latex]\\theta ={60}^{\\circ}[\/latex], which is the angle of the rotation of the axes.<\/p>\n<p id=\"fs-id1167794119112\">To determine the rotated coefficients, use the formulas given above:<\/p>\n<div id=\"fs-id1167794119115\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill {A}^{\\prime }& =\\hfill & A{\\cos}^{2}\\theta +B\\cos\\theta \\sin\\theta +C{\\sin}^{2}\\theta \\hfill \\\\ & =\\hfill & 13{\\cos}^{2}60+\\left(-6\\sqrt{3}\\right)\\cos60\\sin60+7{\\sin}^{2}60\\hfill \\\\ & =\\hfill & 13{\\left(\\frac{1}{2}\\right)}^{2}-6\\sqrt{3}\\left(\\frac{1}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right)+7{\\left(\\frac{\\sqrt{3}}{2}\\right)}^{2}\\hfill \\\\ & =\\hfill & 4,\\hfill \\\\ \\hfill {B}^{\\prime }& =\\hfill & 0,\\hfill \\\\ \\hfill {C}^{\\prime }& =\\hfill & A{\\sin}^{2}\\theta -B\\sin\\theta \\cos\\theta +C{\\cos}^{2}\\theta \\hfill \\\\ & =\\hfill & 13{\\sin}^{2}60+\\left(-6\\sqrt{3}\\right)\\sin60\\cos60=7{\\cos}^{2}60\\hfill \\\\ \\hfill & =\\hfill & {\\left(\\frac{\\sqrt{3}}{2}\\right)}^{2}+6\\sqrt{3}\\left(\\frac{\\sqrt{3}}{2}\\right)\\left(\\frac{1}{2}\\right)+7{\\left(\\frac{1}{2}\\right)}^{2}\\hfill \\\\ & =\\hfill & 16,\\hfill \\\\ \\hfill {D}^{\\prime }& =\\hfill & D\\cos\\theta +E\\sin\\theta \\hfill \\\\ & =\\hfill & \\left(0\\right)\\cos60+\\left(0\\right)\\sin60\\hfill \\\\ & =\\hfill & 0,\\hfill \\\\ \\hfill {E}^{\\prime }& =\\hfill & -D\\sin\\theta +E\\cos\\theta \\hfill \\\\ & =\\hfill & -\\left(0\\right)\\sin60+\\left(0\\right)\\cos60\\hfill \\\\ & =\\hfill & 0,\\hfill \\\\ \\hfill {F}^{\\prime }& =\\hfill & F\\hfill \\\\ & =\\hfill & -256.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167794136912\">The equation of the conic in the rotated coordinate system becomes<\/p>\n<div id=\"fs-id1167794136915\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{}\\\\ \\hfill 4{\\left({x}^{\\prime }\\right)}^{2}+16{\\left({y}^{\\prime }\\right)}^{2}& =\\hfill & 256\\hfill \\\\ \\hfill \\frac{{\\left({x}^{\\prime }\\right)}^{2}}{64}+\\frac{{\\left({y}^{\\prime }\\right)}^{2}}{16}& =\\hfill & 1.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167793389848\">A graph of this conic section appears as follows.<\/p>\n<figure id=\"CNX_Calc_Figure_11_05_020\"><figcaption><\/figcaption><div style=\"width: 429px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225415\/CNX_Calc_Figure_11_05_020.jpg\" alt=\"Graph of an ellipse with equation 13x2 \u2013 6 times the square root of 3 times xy + 7y2 \u2013 256 = 0. The center is at the origin, and the ellipse appears to be skewed 60 degrees. There are dashed red lines along the major and minor axes.\" width=\"419\" height=\"422\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 20. Graph of the ellipse described by the equation [latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}-256=0[\/latex]. The axes are rotated [latex]60^\\circ[\/latex]. The red dashed lines indicate the rotated axes.<\/p>\n<\/div>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Identifying a Rotated Conic.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AF_0mpKk4-Y?controls=0&amp;start=1939&amp;end=2314&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.5ConicSections1939to2314_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;7.5 Conic Sections&#8221; here (opens in new window)<\/a>.<\/p>\n<div id=\"fs-id1167793389930\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1167793389935\" data-type=\"exercise\">\n<div id=\"fs-id1167793389937\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1167793389937\" data-type=\"problem\">\n<p id=\"fs-id1167793389939\">Identify the conic and calculate the angle of rotation of axes for the curve described by the equation<\/p>\n<div id=\"fs-id1167793389943\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]3{x}^{2}+5xy - 2{y}^{2}-125=0[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558819\">Hint<\/span><\/p>\n<div id=\"q44558819\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793390007\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167793390014\">Follow steps 1 and 2 of the five-step method outlined above.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558829\">Show Solution<\/span><\/p>\n<div id=\"q44558829\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793389988\" data-type=\"solution\">\n<p id=\"fs-id1167793389990\">The conic is a hyperbola and the angle of rotation of the axes is [latex]\\theta =22.5^{\\circ }[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm175311\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=175311&theme=oea&iframe_resize_id=ohm175311&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1167793390022\" class=\"key-concepts\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\"><\/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-1855\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>7.5 Conic Sections. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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