{"id":248,"date":"2025-02-13T22:45:33","date_gmt":"2025-02-13T22:45:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-ellipse\/"},"modified":"2026-04-01T09:00:46","modified_gmt":"2026-04-01T09:00:46","slug":"the-ellipse","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-ellipse\/","title":{"raw":"Ellipses: Learn It 1","rendered":"Ellipses: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write equations of ellipses in standard form.<\/li>\r\n \t<li>Graph ellipses.<\/li>\r\n \t<li>Solve applied problems involving ellipses.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Ellipses<\/h2>\r\n<\/div>\r\nCan 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.\u00a0is such a room.[footnote]Architect of the Capitol, accessed: April 15, 2014, http:\/\/www.aoc.gov\/.[\/footnote]\u00a0It 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.\r\n<h3>Writing Equations of Ellipses in Standard Form<\/h3>\r\nA conic section, or <strong>conic<\/strong>, 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.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182705\/CNX_Precalc_Figure_10_01_0022.jpg\" alt=\"\" width=\"976\" height=\"441\" \/>\r\n\r\nConic 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 <strong>ellipse<\/strong> 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 <strong>focus<\/strong> (plural: <strong>foci<\/strong>).\r\n\r\nWe 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.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182707\/CNX_Precalc_Figure_10_01_0032.jpg\" alt=\"\" width=\"487\" height=\"560\" \/>\r\n\r\nEvery ellipse has two axes of symmetry. The longer axis is called the <strong>major axis<\/strong>, and the shorter axis is called the <strong>minor axis<\/strong>. Each endpoint of the major axis is the <strong>vertex<\/strong> of the ellipse (plural: <strong>vertices<\/strong>), and each endpoint of the minor axis is a <strong>co-vertex<\/strong> of the ellipse. The <strong>center of an ellipse<\/strong> 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.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182709\/CNX_Precalc_Figure_10_01_0042.jpg\" alt=\"\" width=\"731\" height=\"366\" \/>\r\n\r\nIn 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.\r\n\r\nTo 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.\r\n<h3>Writing Equations of Ellipses Centered at the Origin in Standard Form<\/h3>\r\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.\r\n\r\nThe key features of the <strong>ellipse<\/strong> are its center, <strong>vertices<\/strong>, <strong>co-vertices<\/strong>, <strong>foci<\/strong>, and lengths and positions of the <strong>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.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>standard form of an ellipse<\/h3>\r\nThe 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\r\n<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/p>\r\nwhere\r\n<ul>\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(\\pm a,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,\\pm b\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are [latex]\\left(\\pm c,0\\right)[\/latex] , where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\r\n<\/ul>\r\nThe 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\r\n<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1[\/latex]<\/p>\r\nwhere\r\n<ul>\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,\\pm a\\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(\\pm b,0\\right)[\/latex]<\/li>\r\n \t<li>the coordinates of the foci are [latex]\\left(0,\\pm c\\right)[\/latex] , where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\r\n<\/ul>\r\nNote 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.\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/09\/Screen-Shot-2015-09-15-at-2.21.34-PM.png\"><img class=\"wp-image-11282 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182713\/Screen-Shot-2015-09-15-at-2.21.34-PM.png\" alt=\"Horizontal ellipse with center (0,0)(b) Vertical ellipse with center(0,0)\" width=\"668\" height=\"327\" \/><\/a>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.<\/strong>\r\n<ol>\r\n \t<li>Determine whether the major axis lies on the <em>x<\/em>- or <em>y<\/em>-axis.\r\n<ol>\r\n \t<li>If the given coordinates of the vertices and foci have the form [latex]\\left(\\pm a,0\\right)[\/latex] and [latex]\\left(\\pm c,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,\\pm a\\right)[\/latex] and [latex]\\left(\\pm c,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<\/section><section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the ellipse that has vertices [latex]\\left(\\pm 8,0\\right)[\/latex] and foci [latex]\\left(\\pm 5,0\\right)?[\/latex][reveal-answer q=\"478058\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"478058\"]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\r\n<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/p>\r\nThe vertices are [latex]\\left(\\pm 8,0\\right)[\/latex], so [latex]a=8[\/latex] and [latex]{a}^{2}=64[\/latex].\r\n\r\nThe foci are [latex]\\left(\\pm 5,0\\right)[\/latex], so [latex]c=5[\/latex] and [latex]{c}^{2}=25[\/latex].\r\n\r\nWe 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:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{c}^{2}={a}^{2}-{b}^{2} \\\\ &amp;25=64-{b}^{2}&amp;&amp; \\text{Substitute for }{c}^{2}\\text{ and }{a}^{2}. \\\\ &amp;{b}^{2}=39&amp;&amp; \\text{Solve for }{b}^{2}. \\end{align}[\/latex]<\/p>\r\nNow 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">What is the standard form equation of the ellipse that has vertices [latex]\\left(0,\\pm 4\\right)[\/latex] and foci [latex]\\left(0,\\pm \\sqrt{15}\\right)?[\/latex][reveal-answer q=\"757761\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"757761\"][latex]{x}^{2}+\\frac{{y}^{2}}{16}=1[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]173939[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form [latex]\\left(\\pm a,0\\right)[\/latex] or [latex]\\left(0,\\pm a\\right)[\/latex]. Similarly, the coordinates of the foci will always have the form [latex]\\left(\\pm c,0\\right)[\/latex] or [latex]\\left(0,\\pm c\\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].<\/section><\/section>\r\n<dl id=\"fs-id1350562\" class=\"definition\">\r\n \t<dd id=\"fs-id1350566\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write equations of ellipses in standard form.<\/li>\n<li>Graph ellipses.<\/li>\n<li>Solve applied problems involving ellipses.<\/li>\n<\/ul>\n<\/section>\n<h2>Ellipses<\/h2>\n<\/div>\n<p>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.\u00a0is such a room.<a class=\"footnote\" title=\"Architect of the Capitol, accessed: April 15, 2014, http:\/\/www.aoc.gov\/.\" id=\"return-footnote-248-1\" href=\"#footnote-248-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0It 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<h3>Writing Equations of Ellipses in Standard Form<\/h3>\n<p>A conic section, or <strong>conic<\/strong>, 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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182705\/CNX_Precalc_Figure_10_01_0022.jpg\" alt=\"\" width=\"976\" height=\"441\" \/><\/p>\n<p>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 <strong>ellipse<\/strong> 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 <strong>focus<\/strong> (plural: <strong>foci<\/strong>).<\/p>\n<p>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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182707\/CNX_Precalc_Figure_10_01_0032.jpg\" alt=\"\" width=\"487\" height=\"560\" \/><\/p>\n<p>Every ellipse has two axes of symmetry. The longer axis is called the <strong>major axis<\/strong>, and the shorter axis is called the <strong>minor axis<\/strong>. Each endpoint of the major axis is the <strong>vertex<\/strong> of the ellipse (plural: <strong>vertices<\/strong>), and each endpoint of the minor axis is a <strong>co-vertex<\/strong> of the ellipse. The <strong>center of an ellipse<\/strong> 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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182709\/CNX_Precalc_Figure_10_01_0042.jpg\" alt=\"\" width=\"731\" height=\"366\" \/><\/p>\n<p>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>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<h3>Writing Equations of Ellipses Centered at the Origin in Standard Form<\/h3>\n<p>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>The key features of the <strong>ellipse<\/strong> are its center, <strong>vertices<\/strong>, <strong>co-vertices<\/strong>, <strong>foci<\/strong>, and lengths and positions of the <strong>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.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>standard form of an ellipse<\/h3>\n<p>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<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/p>\n<p>where<\/p>\n<ul>\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(\\pm a,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,\\pm b\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are [latex]\\left(\\pm c,0\\right)[\/latex] , where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\n<\/ul>\n<p>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<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}}{{b}^{2}}+\\frac{{y}^{2}}{{a}^{2}}=1[\/latex]<\/p>\n<p>where<\/p>\n<ul>\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,\\pm a\\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(\\pm b,0\\right)[\/latex]<\/li>\n<li>the coordinates of the foci are [latex]\\left(0,\\pm c\\right)[\/latex] , where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\n<\/ul>\n<p>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<p><a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/09\/Screen-Shot-2015-09-15-at-2.21.34-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-11282 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182713\/Screen-Shot-2015-09-15-at-2.21.34-PM.png\" alt=\"Horizontal ellipse with center (0,0)(b) Vertical ellipse with center(0,0)\" width=\"668\" height=\"327\" \/><\/a><\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.<\/strong><\/p>\n<ol>\n<li>Determine whether the major axis lies on the <em>x<\/em>&#8211; or <em>y<\/em>-axis.\n<ol>\n<li>If the given coordinates of the vertices and foci have the form [latex]\\left(\\pm a,0\\right)[\/latex] and [latex]\\left(\\pm c,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,\\pm a\\right)[\/latex] and [latex]\\left(\\pm c,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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the ellipse that has vertices [latex]\\left(\\pm 8,0\\right)[\/latex] and foci [latex]\\left(\\pm 5,0\\right)?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q478058\">Show Solution<\/button><\/p>\n<div id=\"q478058\" class=\"hidden-answer\" style=\"display: none\">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<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/p>\n<p>The vertices are [latex]\\left(\\pm 8,0\\right)[\/latex], so [latex]a=8[\/latex] and [latex]{a}^{2}=64[\/latex].<\/p>\n<p>The foci are [latex]\\left(\\pm 5,0\\right)[\/latex], so [latex]c=5[\/latex] and [latex]{c}^{2}=25[\/latex].<\/p>\n<p>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<p style=\"text-align: center;\">[latex]\\begin{align}&{c}^{2}={a}^{2}-{b}^{2} \\\\ &25=64-{b}^{2}&& \\text{Substitute for }{c}^{2}\\text{ and }{a}^{2}. \\\\ &{b}^{2}=39&& \\text{Solve for }{b}^{2}. \\end{align}[\/latex]<\/p>\n<p>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].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">What is the standard form equation of the ellipse that has vertices [latex]\\left(0,\\pm 4\\right)[\/latex] and foci [latex]\\left(0,\\pm \\sqrt{15}\\right)?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q757761\">Show Solution<\/button><\/p>\n<div id=\"q757761\" class=\"hidden-answer\" style=\"display: none\">[latex]{x}^{2}+\\frac{{y}^{2}}{16}=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm173939\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173939&theme=lumen&iframe_resize_id=ohm173939&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form [latex]\\left(\\pm a,0\\right)[\/latex] or [latex]\\left(0,\\pm a\\right)[\/latex]. Similarly, the coordinates of the foci will always have the form [latex]\\left(\\pm c,0\\right)[\/latex] or [latex]\\left(0,\\pm c\\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].<\/section>\n<\/section>\n<dl id=\"fs-id1350562\" class=\"definition\">\n<dd id=\"fs-id1350566\"><\/dd>\n<\/dl>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-248-1\">Architect of the Capitol, accessed: April 15, 2014, http:\/\/www.aoc.gov\/. <a href=\"#return-footnote-248-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":6,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/248"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/248\/revisions"}],"predecessor-version":[{"id":6103,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/248\/revisions\/6103"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/248\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=248"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=248"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=248"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}