{"id":421,"date":"2019-07-15T22:44:54","date_gmt":"2019-07-15T22:44:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/summary-the-ellipse\/"},"modified":"2019-07-15T22:44:54","modified_gmt":"2019-07-15T22:44:54","slug":"summary-the-ellipse","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/summary-the-ellipse\/","title":{"raw":"Summary: The Ellipse","rendered":"Summary: The Ellipse"},"content":{"raw":"\n<h2>Key Equations<\/h2>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td>Horizontal ellipse, center at origin<\/td>\n<td>[latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Vertical ellipse, center at origin<\/td>\n<td>[latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Horizontal ellipse, center [latex]\\left(h,k\\right)[\/latex]<\/td>\n<td>[latex]\\dfrac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\dfrac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Vertical ellipse, center [latex]\\left(h,k\\right)[\/latex]<\/td>\n<td>[latex]\\dfrac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\dfrac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\text{ }a&gt;b[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>An ellipse is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).<\/li>\n \t<li>When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form.<\/li>\n \t<li>When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse.<\/li>\n \t<li>When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse.<\/li>\n \t<li>Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<strong>center of an ellipse<\/strong> the midpoint of both the major and minor axes\n\n<strong>conic section<\/strong> any shape resulting from the intersection of a right circular cone with a plane\n\n<strong>ellipse<\/strong> 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\n\n<strong>foci<\/strong> plural of focus\n\n<strong>focus (of an ellipse)<\/strong> one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point [latex]\\left(x,y\\right)[\/latex] on the ellipse is a constant\n\n<strong>major axis<\/strong> the longer of the two axes of an ellipse\n\n<strong>minor axis<\/strong> the shorter of the two axes of an ellipse\n","rendered":"<h2>Key Equations<\/h2>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td>Horizontal ellipse, center at origin<\/td>\n<td>[latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Vertical ellipse, center at origin<\/td>\n<td>[latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Horizontal ellipse, center [latex]\\left(h,k\\right)[\/latex]<\/td>\n<td>[latex]\\dfrac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\dfrac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Vertical ellipse, center [latex]\\left(h,k\\right)[\/latex]<\/td>\n<td>[latex]\\dfrac{{\\left(x-h\\right)}^{2}}{{b}^{2}}+\\dfrac{{\\left(y-k\\right)}^{2}}{{a}^{2}}=1,\\text{ }a>b[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>An ellipse is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).<\/li>\n<li>When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form.<\/li>\n<li>When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse.<\/li>\n<li>When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse.<\/li>\n<li>Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>center of an ellipse<\/strong> the midpoint of both the major and minor axes<\/p>\n<p><strong>conic section<\/strong> any shape resulting from the intersection of a right circular cone with a plane<\/p>\n<p><strong>ellipse<\/strong> 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<\/p>\n<p><strong>foci<\/strong> plural of focus<\/p>\n<p><strong>focus (of an ellipse)<\/strong> one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point [latex]\\left(x,y\\right)[\/latex] on the ellipse is a constant<\/p>\n<p><strong>major axis<\/strong> the longer of the two axes of an ellipse<\/p>\n<p><strong>minor axis<\/strong> the shorter of the two axes of an ellipse<\/p>\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-421\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":8,"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\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"}]","CANDELA_OUTCOMES_GUID":"2abc1bac-b41b-40ae-bbb4-7d6f04adc34d","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-421","chapter","type-chapter","status-publish","hentry"],"part":413,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/421","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/421\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/parts\/413"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/421\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/media?parent=421"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=421"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/contributor?post=421"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/license?post=421"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}