{"id":3300,"date":"2025-08-16T01:53:09","date_gmt":"2025-08-16T01:53:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3300"},"modified":"2026-07-13T18:00:50","modified_gmt":"2026-07-13T18:00:50","slug":"conic-sections-in-polar-coordinates-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/conic-sections-in-polar-coordinates-apply-it\/","title":{"raw":"Conic Sections in Polar Coordinates: Apply It","rendered":"Conic Sections in Polar Coordinates: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify a conic in polar form.<\/li>\r\n \t<li>Graph the polar equations of conics.<\/li>\r\n \t<li>De\ufb01ne conics in terms of a focus and a directrix.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">A conic section in polar form with focus at the origin has the equation [latex]r = \\frac{ep}{1 \\pm e\\cos\\theta}[\/latex] or [latex]r = \\frac{ep}{1 \\pm e\\sin\\theta}[\/latex], where [latex]e[\/latex] is the eccentricity and [latex]p[\/latex] is the distance to the directrix (a fixed reference line). The eccentricity determines the shape:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]0 \\leq e &lt; 1[\/latex]: ellipse<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]e = 1[\/latex]: parabola<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]e &gt; 1[\/latex]: hyperbola<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The trigonometric function indicates directrix orientation: [latex]\\cos\\theta[\/latex] means a vertical directrix ([latex]x = \\pm p[\/latex]), while [latex]\\sin\\theta[\/latex] means a horizontal directrix ([latex]y = \\pm p[\/latex]).<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Orbital Mechanics<\/h2>\r\n<p class=\"whitespace-normal break-words\">Satellites, planets, and comets follow conic paths around celestial bodies. The shape of an orbit depends on the object's velocity and distance from the body it orbits. Understanding these polar equations helps scientists predict orbital behavior and plan space missions.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A satellite's orbital path is described by [latex]r = \\frac{6}{3 + 2\\sin\\theta}[\/latex], where [latex]r[\/latex] is measured in thousands of kilometers. Identify the type of orbit, the eccentricity, and the directrix.<\/p>\r\n<p class=\"whitespace-normal break-words\">[reveal-answer q=\"681883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"681883\"]First, rewrite in standard form by multiplying numerator and denominator by [latex]\\frac{1}{3}[\/latex]: [latex]\\begin{aligned} r &amp;= \\frac{6}{3 + 2\\sin\\theta} \\cdot \\frac{\\frac{1}{3}}{\\frac{1}{3}} \\ &amp;= \\frac{6 \\cdot \\frac{1}{3}}{3 \\cdot \\frac{1}{3} + 2 \\cdot \\frac{1}{3}\\sin\\theta} \\ &amp;= \\frac{2}{1 + \\frac{2}{3}\\sin\\theta} \\end{aligned}[\/latex] Now identify the characteristics: The eccentricity is [latex]e = \\frac{2}{3}[\/latex]. Since [latex]e &lt; 1[\/latex], this is an ellipse. Since [latex]\\sin\\theta[\/latex] appears in the denominator with a plus sign, the directrix is [latex]y = p[\/latex] where [latex]p &gt; 0[\/latex]. To find [latex]p[\/latex], use [latex]ep = 2[\/latex]: [latex]\\begin{aligned} \\frac{2}{3} \\cdot p &amp;= 2 \\ p &amp;= 2 \\cdot \\frac{3}{2} \\ p &amp;= 3 \\end{aligned}[\/latex] The satellite follows an elliptical orbit with eccentricity [latex]e = \\frac{2}{3}[\/latex] and directrix [latex]y = 3[\/latex] (3,000 km above the origin).[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A space probe follows the path [latex]r = \\frac{7}{2 - 2\\sin\\theta}[\/latex]. Identify the orbit characteristics.<\/p>\r\n<p class=\"whitespace-normal break-words\">[reveal-answer q=\"109528\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"109528\"]Multiply numerator and denominator by [latex]\\frac{1}{2}[\/latex]: [latex]\\begin{aligned} r &amp;= \\frac{7}{2 - 2\\sin\\theta} \\cdot \\frac{\\frac{1}{2}}{\\frac{1}{2}} \\ &amp;= \\frac{\\frac{7}{2}}{1 - \\sin\\theta} \\end{aligned}[\/latex] The eccentricity is [latex]e = 1[\/latex], so this is a parabola. Since [latex]\\sin\\theta[\/latex] appears with a minus sign, the directrix is [latex]y = -p[\/latex] where [latex]p &gt; 0[\/latex]. From [latex]ep = \\frac{7}{2}[\/latex]: [latex]\\begin{aligned} (1) \\cdot p &amp;= \\frac{7}{2} \\ p &amp;= \\frac{7}{2} \\end{aligned}[\/latex] The directrix is [latex]y = -\\frac{7}{2} = -3.5[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox connectIt\" aria-label=\"Connect It\">A parabolic orbit ([latex]e = 1[\/latex]) represents the boundary between closed orbits (ellipses) and open orbits (hyperbolas). Objects at exactly escape velocity follow parabolic paths!<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]327161[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]327162[\/ohm_question]<\/section><section class=\"textbox recall\" aria-label=\"Recall\">The general forms are [latex]r = \\frac{ep}{1 \\pm e\\cos\\theta}[\/latex] or [latex]r = \\frac{ep}{1 \\pm e\\sin\\theta}[\/latex]. Choose [latex]\\cos\\theta[\/latex] for a vertical directrix ([latex]x = \\pm p[\/latex]) and [latex]\\sin\\theta[\/latex] for a horizontal directrix ([latex]y = \\pm p[\/latex]). Use [latex]+[\/latex] when [latex]p &gt; 0[\/latex] and [latex]-[\/latex] when [latex]p &lt; 0[\/latex].<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Mission control needs to place a satellite in an elliptical orbit with focus at Earth's center, eccentricity [latex]e = \\frac{3}{5}[\/latex], and directrix at [latex]x = 4[\/latex] (in thousands of km). Write the polar equation for this orbit.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Solution:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The directrix [latex]x = 4[\/latex] tells us:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Use [latex]\\cos\\theta[\/latex] in the denominator (vertical directrix)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Since [latex]4 &gt; 0[\/latex], use the [latex]+[\/latex] sign<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]p = 4[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The standard form is:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]r = \\frac{ep}{1 + e\\cos\\theta}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Substituting [latex]e = \\frac{3}{5}[\/latex] and [latex]p = 4[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} r &amp;= \\frac{\\frac{3}{5} \\cdot 4}{1 + \\frac{3}{5}\\cos\\theta} \\ &amp;= \\frac{\\frac{12}{5}}{1 + \\frac{3}{5}\\cos\\theta} \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">To eliminate the fraction in the numerator, multiply by [latex]\\frac{5}{5}[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} r &amp;= \\frac{\\frac{12}{5}}{\\frac{5}{5} + \\frac{3}{5}\\cos\\theta} \\ &amp;= \\frac{12}{5} \\cdot \\frac{5}{5 + 3\\cos\\theta} \\ &amp;= \\frac{12}{5 + 3\\cos\\theta} \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The orbital equation is [latex]r = \\frac{12}{5 + 3\\cos\\theta}[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox youChoose\" aria-label=\"You Choose\">\r\n<p class=\"whitespace-normal break-words\">[choosedataset divId=\"tnh-choose-dataset\" title=\"Choose Your Own Orbit\" label=\"\" default=\"Choose an Orbit\"]\r\n[datasetoption]\r\n[displayname]Communications Satellite[\/displayname]\r\n[ohmid]59200[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Interstellar Probe[\/displayname]\r\n[ohmid]59201[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Solar Observation Probe[\/displayname]\r\n[ohmid]59202[\/ohmid]\r\n[\/datasetoption]\r\n[\/choosedataset]<\/p>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify a conic in polar form.<\/li>\n<li>Graph the polar equations of conics.<\/li>\n<li>De\ufb01ne conics in terms of a focus and a directrix.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">A conic section in polar form with focus at the origin has the equation [latex]r = \\frac{ep}{1 \\pm e\\cos\\theta}[\/latex] or [latex]r = \\frac{ep}{1 \\pm e\\sin\\theta}[\/latex], where [latex]e[\/latex] is the eccentricity and [latex]p[\/latex] is the distance to the directrix (a fixed reference line). The eccentricity determines the shape:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">If [latex]0 \\leq e < 1[\/latex]: ellipse<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]e = 1[\/latex]: parabola<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]e > 1[\/latex]: hyperbola<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The trigonometric function indicates directrix orientation: [latex]\\cos\\theta[\/latex] means a vertical directrix ([latex]x = \\pm p[\/latex]), while [latex]\\sin\\theta[\/latex] means a horizontal directrix ([latex]y = \\pm p[\/latex]).<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Orbital Mechanics<\/h2>\n<p class=\"whitespace-normal break-words\">Satellites, planets, and comets follow conic paths around celestial bodies. The shape of an orbit depends on the object&#8217;s velocity and distance from the body it orbits. Understanding these polar equations helps scientists predict orbital behavior and plan space missions.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A satellite&#8217;s orbital path is described by [latex]r = \\frac{6}{3 + 2\\sin\\theta}[\/latex], where [latex]r[\/latex] is measured in thousands of kilometers. Identify the type of orbit, the eccentricity, and the directrix.<\/p>\n<p class=\"whitespace-normal break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q681883\">Show Solution<\/button><\/p>\n<div id=\"q681883\" class=\"hidden-answer\" style=\"display: none\">First, rewrite in standard form by multiplying numerator and denominator by [latex]\\frac{1}{3}[\/latex]: [latex]\\begin{aligned} r &= \\frac{6}{3 + 2\\sin\\theta} \\cdot \\frac{\\frac{1}{3}}{\\frac{1}{3}} \\ &= \\frac{6 \\cdot \\frac{1}{3}}{3 \\cdot \\frac{1}{3} + 2 \\cdot \\frac{1}{3}\\sin\\theta} \\ &= \\frac{2}{1 + \\frac{2}{3}\\sin\\theta} \\end{aligned}[\/latex] Now identify the characteristics: The eccentricity is [latex]e = \\frac{2}{3}[\/latex]. Since [latex]e < 1[\/latex], this is an ellipse. Since [latex]\\sin\\theta[\/latex] appears in the denominator with a plus sign, the directrix is [latex]y = p[\/latex] where [latex]p > 0[\/latex]. To find [latex]p[\/latex], use [latex]ep = 2[\/latex]: [latex]\\begin{aligned} \\frac{2}{3} \\cdot p &= 2 \\ p &= 2 \\cdot \\frac{3}{2} \\ p &= 3 \\end{aligned}[\/latex] The satellite follows an elliptical orbit with eccentricity [latex]e = \\frac{2}{3}[\/latex] and directrix [latex]y = 3[\/latex] (3,000 km above the origin).<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A space probe follows the path [latex]r = \\frac{7}{2 - 2\\sin\\theta}[\/latex]. Identify the orbit characteristics.<\/p>\n<p class=\"whitespace-normal break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q109528\">Show Solution<\/button><\/p>\n<div id=\"q109528\" class=\"hidden-answer\" style=\"display: none\">Multiply numerator and denominator by [latex]\\frac{1}{2}[\/latex]: [latex]\\begin{aligned} r &= \\frac{7}{2 - 2\\sin\\theta} \\cdot \\frac{\\frac{1}{2}}{\\frac{1}{2}} \\ &= \\frac{\\frac{7}{2}}{1 - \\sin\\theta} \\end{aligned}[\/latex] The eccentricity is [latex]e = 1[\/latex], so this is a parabola. Since [latex]\\sin\\theta[\/latex] appears with a minus sign, the directrix is [latex]y = -p[\/latex] where [latex]p > 0[\/latex]. From [latex]ep = \\frac{7}{2}[\/latex]: [latex]\\begin{aligned} (1) \\cdot p &= \\frac{7}{2} \\ p &= \\frac{7}{2} \\end{aligned}[\/latex] The directrix is [latex]y = -\\frac{7}{2} = -3.5[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">A parabolic orbit ([latex]e = 1[\/latex]) represents the boundary between closed orbits (ellipses) and open orbits (hyperbolas). Objects at exactly escape velocity follow parabolic paths!<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm327161\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=327161&theme=lumen&iframe_resize_id=ohm327161&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm327162\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=327162&theme=lumen&iframe_resize_id=ohm327162&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">The general forms are [latex]r = \\frac{ep}{1 \\pm e\\cos\\theta}[\/latex] or [latex]r = \\frac{ep}{1 \\pm e\\sin\\theta}[\/latex]. Choose [latex]\\cos\\theta[\/latex] for a vertical directrix ([latex]x = \\pm p[\/latex]) and [latex]\\sin\\theta[\/latex] for a horizontal directrix ([latex]y = \\pm p[\/latex]). Use [latex]+[\/latex] when [latex]p > 0[\/latex] and [latex]-[\/latex] when [latex]p < 0[\/latex].<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Mission control needs to place a satellite in an elliptical orbit with focus at Earth&#8217;s center, eccentricity [latex]e = \\frac{3}{5}[\/latex], and directrix at [latex]x = 4[\/latex] (in thousands of km). Write the polar equation for this orbit.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Solution:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The directrix [latex]x = 4[\/latex] tells us:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Use [latex]\\cos\\theta[\/latex] in the denominator (vertical directrix)<\/li>\n<li class=\"whitespace-normal break-words\">Since [latex]4 > 0[\/latex], use the [latex]+[\/latex] sign<\/li>\n<li class=\"whitespace-normal break-words\">[latex]p = 4[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The standard form is:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]r = \\frac{ep}{1 + e\\cos\\theta}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Substituting [latex]e = \\frac{3}{5}[\/latex] and [latex]p = 4[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} r &= \\frac{\\frac{3}{5} \\cdot 4}{1 + \\frac{3}{5}\\cos\\theta} \\ &= \\frac{\\frac{12}{5}}{1 + \\frac{3}{5}\\cos\\theta} \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">To eliminate the fraction in the numerator, multiply by [latex]\\frac{5}{5}[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} r &= \\frac{\\frac{12}{5}}{\\frac{5}{5} + \\frac{3}{5}\\cos\\theta} \\ &= \\frac{12}{5} \\cdot \\frac{5}{5 + 3\\cos\\theta} \\ &= \\frac{12}{5 + 3\\cos\\theta} \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The orbital equation is [latex]r = \\frac{12}{5 + 3\\cos\\theta}[\/latex].<\/p>\n<\/section>\n<section class=\"textbox youChoose\" aria-label=\"You Choose\">\n<p class=\"whitespace-normal break-words\">\n<div id=\"tnh-choose-dataset\" class=\"chooseDataset\">\n<h3>Choose Your Own Orbit<\/h3>\n<form><select name=\"dataset\"><option value=\"\">Choose an Orbit<\/option><option value=\"59200\">Communications Satellite<\/option><option value=\"59201\">Interstellar Probe<\/option><option value=\"59202\">Solar Observation Probe<\/option><\/select><\/form>\n<div class=\"ohmContainer\"><\/div>\n<\/p><\/div>\n<\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":31,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":[{"divId":"tnh-choose-dataset","title":"Choose Your Own Orbit","label":"","default":"Choose an Orbit","try_it_collection":[{"displayName":"Communications Satellite","value":"59200"},{"displayName":"Interstellar Probe","value":"59201"},{"displayName":"Solar Observation Probe","value":"59202"}]}],"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3300"}],"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\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3300\/revisions"}],"predecessor-version":[{"id":6291,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3300\/revisions\/6291"}],"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\/3300\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3300"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3300"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3300"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}