{"id":1170,"date":"2021-11-11T17:37:26","date_gmt":"2021-11-11T17:37:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-conic-sections\/"},"modified":"2022-10-20T23:40:15","modified_gmt":"2022-10-20T23:40:15","slug":"summary-of-conic-sections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-conic-sections\/","title":{"raw":"Summary of Conic Sections","rendered":"Summary of Conic Sections"},"content":{"raw":"<section id=\"fs-id1167793390022\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167793390029\" data-bullet-style=\"bullet\">\r\n \t<li>The equation of a vertical parabola in standard form with given focus and directrix is [latex]y=\\frac{1}{4p}{\\left(x-h\\right)}^{2}+k[\/latex] where <em data-effect=\"italics\">p<\/em> is the distance from the vertex to the focus and [latex]\\left(h,k\\right)[\/latex] are the coordinates of the vertex.<\/li>\r\n \t<li>The equation of a horizontal ellipse in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the major axis has length 2<em data-effect=\"italics\">a,<\/em> the minor axis has length 2<em data-effect=\"italics\">b<\/em>, and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\r\n \t<li>The equation of a horizontal hyperbola in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the vertices are located at [latex]\\left(h\\pm a,k\\right)[\/latex], and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/li>\r\n \t<li>The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.<\/li>\r\n \t<li>The polar equation of a conic section with eccentricity <em data-effect=\"italics\">e<\/em> is [latex]r=\\frac{ep}{1\\pm e\\cos\\theta }[\/latex] or [latex]r=\\frac{ep}{1\\pm e\\sin\\theta }[\/latex], where <em data-effect=\"italics\">p<\/em> represents the focal parameter.<\/li>\r\n \t<li>To identify a conic generated by the equation [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], first calculate the discriminant [latex]D=4AC-{B}^{2}[\/latex]. If [latex]D&gt;0[\/latex] then the conic is an ellipse, if [latex]D=0[\/latex] then the conic is a parabola, and if [latex]D&lt;0[\/latex] then the conic is a hyperbola.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1167794101792\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167794049400\">\r\n \t<dt>conic section<\/dt>\r\n \t<dd id=\"fs-id1167794049405\">a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049410\">\r\n \t<dt>directrix<\/dt>\r\n \t<dd id=\"fs-id1167794049415\">a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049421\">\r\n \t<dt>discriminant<\/dt>\r\n \t<dd id=\"fs-id1167794049426\">the value [latex]4AC-{B}^{2}[\/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[\/latex], is called a discriminant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049471\">\r\n \t<dt>eccentricity<\/dt>\r\n \t<dd id=\"fs-id1167794049476\">the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049482\">\r\n \t<dt>focal parameter<\/dt>\r\n \t<dd id=\"fs-id1167794049487\">the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049460\">\r\n \t<dt>focus<\/dt>\r\n \t<dd id=\"fs-id1167794049465\">a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049493\">\r\n \t<dt>general form<\/dt>\r\n \t<dd id=\"fs-id1167794049498\">an equation of a conic section written as a general second-degree equation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049502\">\r\n \t<dt>major axis<\/dt>\r\n \t<dd id=\"fs-id1167794049508\">the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049514\">\r\n \t<dt>minor axis<\/dt>\r\n \t<dd id=\"fs-id1167794049519\">the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049525\">\r\n \t<dt>nappe<\/dt>\r\n \t<dd id=\"fs-id1167794049530\">a nappe is one half of a double cone<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049535\">\r\n \t<dt>standard form<\/dt>\r\n \t<dd id=\"fs-id1167794049540\">an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049546\">\r\n \t<dt>vertex<\/dt>\r\n \t<dd id=\"fs-id1167794049551\">a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1167793390022\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167793390029\" data-bullet-style=\"bullet\">\n<li>The equation of a vertical parabola in standard form with given focus and directrix is [latex]y=\\frac{1}{4p}{\\left(x-h\\right)}^{2}+k[\/latex] where <em data-effect=\"italics\">p<\/em> is the distance from the vertex to the focus and [latex]\\left(h,k\\right)[\/latex] are the coordinates of the vertex.<\/li>\n<li>The equation of a horizontal ellipse in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the major axis has length 2<em data-effect=\"italics\">a,<\/em> the minor axis has length 2<em data-effect=\"italics\">b<\/em>, and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\n<li>The equation of a horizontal hyperbola in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the vertices are located at [latex]\\left(h\\pm a,k\\right)[\/latex], and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/li>\n<li>The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.<\/li>\n<li>The polar equation of a conic section with eccentricity <em data-effect=\"italics\">e<\/em> is [latex]r=\\frac{ep}{1\\pm e\\cos\\theta }[\/latex] or [latex]r=\\frac{ep}{1\\pm e\\sin\\theta }[\/latex], where <em data-effect=\"italics\">p<\/em> represents the focal parameter.<\/li>\n<li>To identify a conic generated by the equation [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], first calculate the discriminant [latex]D=4AC-{B}^{2}[\/latex]. If [latex]D>0[\/latex] then the conic is an ellipse, if [latex]D=0[\/latex] then the conic is a parabola, and if [latex]D<0[\/latex] then the conic is a hyperbola.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1167794101792\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167794049400\">\n<dt>conic section<\/dt>\n<dd id=\"fs-id1167794049405\">a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049410\">\n<dt>directrix<\/dt>\n<dd id=\"fs-id1167794049415\">a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049421\">\n<dt>discriminant<\/dt>\n<dd id=\"fs-id1167794049426\">the value [latex]4AC-{B}^{2}[\/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[\/latex], is called a discriminant<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049471\">\n<dt>eccentricity<\/dt>\n<dd id=\"fs-id1167794049476\">the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049482\">\n<dt>focal parameter<\/dt>\n<dd id=\"fs-id1167794049487\">the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049460\">\n<dt>focus<\/dt>\n<dd id=\"fs-id1167794049465\">a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049493\">\n<dt>general form<\/dt>\n<dd id=\"fs-id1167794049498\">an equation of a conic section written as a general second-degree equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049502\">\n<dt>major axis<\/dt>\n<dd id=\"fs-id1167794049508\">the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049514\">\n<dt>minor axis<\/dt>\n<dd id=\"fs-id1167794049519\">the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049525\">\n<dt>nappe<\/dt>\n<dd id=\"fs-id1167794049530\">a nappe is one half of a double cone<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049535\">\n<dt>standard form<\/dt>\n<dd id=\"fs-id1167794049540\">an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049546\">\n<dt>vertex<\/dt>\n<dd id=\"fs-id1167794049551\">a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch<\/dd>\n<\/dl>\n<\/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-1170\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <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-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/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-3\/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 <\/section>","protected":false},"author":349141,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1170","chapter","type-chapter","status-publish","hentry"],"part":1150,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1170","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1170\/revisions"}],"predecessor-version":[{"id":3683,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1170\/revisions\/3683"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/1150"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1170\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1170"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1170"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1170"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}