{"id":1834,"date":"2015-11-12T18:30:44","date_gmt":"2015-11-12T18:30:44","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1834"},"modified":"2015-11-12T18:30:44","modified_gmt":"2015-11-12T18:30:44","slug":"deriving-the-equation-of-an-ellipse-centered-at-the-origin","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/deriving-the-equation-of-an-ellipse-centered-at-the-origin\/","title":{"raw":"Deriving the Equation of an Ellipse Centered at the Origin","rendered":"Deriving the Equation of an Ellipse Centered at the Origin"},"content":{"raw":"

To derive the equation of an ellipse<\/strong> 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 Figure 5.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 5<\/b>[\/caption]\n\nIf [latex]\\left(a,0\\right)[\/latex] is a vertex<\/strong> 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 foci<\/strong> to the vertex is\n<\/p>

[latex]\\left(a+c\\right)+\\left(a-c\\right)=2a[\/latex]<\/div>\nIf [latex]\\left(x,y\\right)[\/latex] is a point on the ellipse, then we can define the following variables:\n
[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>\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.\n
[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>\nThus, 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.\n

Writing Equations of Ellipses Centered at the Origin in Standard Form<\/h2>\nStandard 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.\n\nThe key features of the ellipse<\/strong> are its center, vertices<\/strong>, co-vertices<\/strong>, foci<\/strong>, and lengths and positions of the major and minor axes<\/strong>. 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.\n
\n

A General Note: Standard Forms of the Equation of an Ellipse with Center (0,0)<\/h3>\nThe standard form of the equation of an ellipse with center [latex]\\left(0,0\\right)[\/latex] and major axis on the x-axis<\/em> is\n
[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/div>\nwhere\n