{"id":2562,"date":"2024-08-06T23:57:01","date_gmt":"2024-08-06T23:57:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2562"},"modified":"2025-08-15T16:38:58","modified_gmt":"2025-08-15T16:38:58","slug":"ellipses-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/ellipses-learn-it-1\/","title":{"raw":"Ellipses: Learn It 1","rendered":"Ellipses: Learn It 1"},"content":{"raw":"<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, understanding how their position changes based on whether they are centered at the origin or at another point<\/li>\r\n \t<li>Solve applied problems involving ellipses<\/li>\r\n<\/ul>\r\n<\/section>\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"300\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202208\/CNX_Precalc_Figure_10_01_001n2.jpg\" alt=\"A large ornate room with statues.\" width=\"300\" height=\"199\" data-media-type=\"image\/jpg\" \/> <b>Figure 1.<\/b> The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)[\/caption]\r\n<h2>Ellipses<\/h2>\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. <a href=\"http:\/\/www.aoc.gov\" target=\"_blank\" rel=\"noopener\">http:\/\/www.aoc.gov<\/a>. Accessed April 15, 2014.[\/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. The oval-shaped room is called an <strong>ellipse<\/strong>, and it is this unique shape that allows sound to travel along the walls.\r\n\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[caption id=\"\" align=\"aligncenter\" width=\"976\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202211\/CNX_Precalc_Figure_10_01_0022.jpg\" alt=\"\" width=\"976\" height=\"441\" \/> Different types of conic shapes[\/caption]\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>) of the ellipse.\r\n\r\nThe angle at which the plane intersects the cone determines the shape. When the plane intersects the cone at an angle, but not steep enough to reach the base of the cone, the resulting shape is an ellipse. If we pull out the cross section from the figure on the right, we will get an ellipse.\r\n\r\nHere are some key characteristics:\r\n<ul>\r\n \t<li>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>.<\/li>\r\n \t<li>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.<\/li>\r\n \t<li>The <strong>center of an ellipse<\/strong> is the midpoint of both the major and minor axes. The axes are perpendicular at the center.<\/li>\r\n \t<li>The <strong>foci<\/strong> 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.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_2564\" align=\"aligncenter\" width=\"400\"]<img class=\"wp-image-2564\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM.png\" alt=\"\" width=\"400\" height=\"210\" \/> Diagram of an ellipse with key features labeled[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>ellipse<\/h3>\r\nAn <strong>ellipse<\/strong> is a geometric shape that looks like a stretched-out circle.\r\n\r\nAn <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>) of the ellipse.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write equations of ellipses in standard form<\/li>\n<li>Graph ellipses, understanding how their position changes based on whether they are centered at the origin or at another point<\/li>\n<li>Solve applied problems involving ellipses<\/li>\n<\/ul>\n<\/section>\n<figure style=\"width: 300px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202208\/CNX_Precalc_Figure_10_01_001n2.jpg\" alt=\"A large ornate room with statues.\" width=\"300\" height=\"199\" data-media-type=\"image\/jpg\" \/><figcaption class=\"wp-caption-text\"><b>Figure 1.<\/b> The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)<\/figcaption><\/figure>\n<h2>Ellipses<\/h2>\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. http:\/\/www.aoc.gov. Accessed April 15, 2014.\" id=\"return-footnote-2562-1\" href=\"#footnote-2562-1\" 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. The oval-shaped room is called an <strong>ellipse<\/strong>, and it is this unique shape that allows sound to travel along the walls.<\/p>\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<figure style=\"width: 976px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202211\/CNX_Precalc_Figure_10_01_0022.jpg\" alt=\"\" width=\"976\" height=\"441\" \/><figcaption class=\"wp-caption-text\">Different types of conic shapes<\/figcaption><\/figure>\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>) of the ellipse.<\/p>\n<p>The angle at which the plane intersects the cone determines the shape. When the plane intersects the cone at an angle, but not steep enough to reach the base of the cone, the resulting shape is an ellipse. If we pull out the cross section from the figure on the right, we will get an ellipse.<\/p>\n<p>Here are some key characteristics:<\/p>\n<ul>\n<li>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>.<\/li>\n<li>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.<\/li>\n<li>The <strong>center of an ellipse<\/strong> is the midpoint of both the major and minor axes. The axes are perpendicular at the center.<\/li>\n<li>The <strong>foci<\/strong> 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.<\/li>\n<\/ul>\n<figure id=\"attachment_2564\" aria-describedby=\"caption-attachment-2564\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2564\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM.png\" alt=\"\" width=\"400\" height=\"210\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM.png 1032w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM-300x158.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM-1024x538.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM-768x403.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM-65x34.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM-225x118.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/06234133\/Screenshot-2024-08-06-at-4.41.28%E2%80%AFPM-350x184.png 350w\" sizes=\"(max-width: 400px) 100vw, 400px\" \/><figcaption id=\"caption-attachment-2564\" class=\"wp-caption-text\">Diagram of an ellipse with key features labeled<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>ellipse<\/h3>\n<p>An <strong>ellipse<\/strong> is a geometric shape that looks like a stretched-out circle.<\/p>\n<p>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>) of the ellipse.<\/p>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-2562-1\">Architect of the Capitol. <a href=\"http:\/\/www.aoc.gov\" target=\"_blank\" rel=\"noopener\">http:\/\/www.aoc.gov<\/a>. Accessed April 15, 2014. <a href=\"#return-footnote-2562-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":12,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":345,"module-header":"learn_it","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2562"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2562\/revisions"}],"predecessor-version":[{"id":7884,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2562\/revisions\/7884"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/345"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2562\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2562"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2562"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2562"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}