{"id":1603,"date":"2025-07-25T04:03:09","date_gmt":"2025-07-25T04:03:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1603"},"modified":"2026-03-12T07:26:37","modified_gmt":"2026-03-12T07:26:37","slug":"conic-sections-in-polar-coordinates-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/conic-sections-in-polar-coordinates-fresh-take\/","title":{"raw":"Conic Sections in Polar Coordinates: Fresh Take","rendered":"Conic Sections in Polar Coordinates: Fresh Take"},"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>\r\n<h2>Identifying a Conic in Polar Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"62\" data-end=\"208\">Conics can also be expressed in <strong data-start=\"94\" data-end=\"108\">polar form<\/strong>, where the focus is placed at the pole (origin). In this setting, the general form of a conic is:<\/p>\r\n<p data-start=\"62\" data-end=\"208\">\\[\r\nr = \\dfrac{ed}{1 \\pm e\\cos\\theta} \\quad \\text{or} \\quad\r\nr = \\dfrac{ed}{1 \\pm e\\sin\\theta}\r\n\\]<\/p>\r\n<p data-start=\"309\" data-end=\"316\">Here:<\/p>\r\n\r\n<ul data-start=\"317\" data-end=\"567\">\r\n \t<li data-start=\"317\" data-end=\"384\">\r\n<p data-start=\"319\" data-end=\"384\">[latex]e[\/latex] = eccentricity (determines the type of conic).<\/p>\r\n<\/li>\r\n \t<li data-start=\"385\" data-end=\"466\">\r\n<p data-start=\"387\" data-end=\"466\">[latex]d[\/latex] = directrix constant (distance related to conic definition).<\/p>\r\n<\/li>\r\n \t<li data-start=\"467\" data-end=\"567\">\r\n<p data-start=\"469\" data-end=\"567\">The sign\/choice of cosine or sine determines orientation: right\/left (cosine) or up\/down (sine).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: How to Identify the Conic<\/strong>\r\n<ol>\r\n \t<li data-start=\"619\" data-end=\"805\">\r\n<p data-start=\"622\" data-end=\"652\"><strong data-start=\"622\" data-end=\"650\">Look at the Eccentricity<\/strong><\/p>\r\n\r\n<ul data-start=\"656\" data-end=\"805\">\r\n \t<li data-start=\"656\" data-end=\"690\">\r\n<p data-start=\"658\" data-end=\"690\">[latex]e=1[\/latex] \u2192 parabola.<\/p>\r\n<\/li>\r\n \t<li data-start=\"694\" data-end=\"766\">\r\n<p data-start=\"696\" data-end=\"766\">[latex]e&lt;1[\/latex] \u2192 ellipse (if [latex]e=0[\/latex], it\u2019s a circle).<\/p>\r\n<\/li>\r\n \t<li data-start=\"770\" data-end=\"805\">\r\n<p data-start=\"772\" data-end=\"805\">[latex]e&gt;1[\/latex] \u2192 hyperbola.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"807\" data-end=\"1077\">\r\n<p data-start=\"810\" data-end=\"833\"><strong data-start=\"810\" data-end=\"831\">Check Orientation<\/strong><\/p>\r\n\r\n<ul data-start=\"837\" data-end=\"1077\">\r\n \t<li data-start=\"837\" data-end=\"922\">\r\n<p data-start=\"839\" data-end=\"922\">[latex]r=\\dfrac{ed}{1+e\\cos\\theta}[\/latex] \u2192 focus at origin, directrix vertical.<\/p>\r\n<\/li>\r\n \t<li data-start=\"926\" data-end=\"996\">\r\n<p data-start=\"928\" data-end=\"996\">[latex]r=\\dfrac{ed}{1-e\\cos\\theta}[\/latex] \u2192 similar but mirrored.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1000\" data-end=\"1077\">\r\n<p data-start=\"1002\" data-end=\"1077\">[latex]r=\\dfrac{ed}{1\\pm e\\sin\\theta}[\/latex] \u2192 opens upward or downward.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1707\" data-end=\"1869\">\r\n<p data-start=\"1710\" data-end=\"1728\"><strong data-start=\"1710\" data-end=\"1726\">Summary Rule<\/strong><\/p>\r\n\r\n<ul data-start=\"1732\" data-end=\"1869\">\r\n \t<li data-start=\"1732\" data-end=\"1798\">\r\n<p data-start=\"1734\" data-end=\"1798\">The <em data-start=\"1738\" data-end=\"1752\">eccentricity<\/em> [latex]e[\/latex] tells you <strong data-start=\"1780\" data-end=\"1795\">which conic<\/strong>.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1802\" data-end=\"1869\">\r\n<p data-start=\"1804\" data-end=\"1869\">The <em data-start=\"1808\" data-end=\"1837\">function (cos\/sin) and sign<\/em> tell you <strong data-start=\"1847\" data-end=\"1866\">which direction<\/strong>.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Identify the conic represented by each polar equation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]r = \\frac{6}{1 + \\cos\\theta}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]r = \\frac{8}{2 - \\sin\\theta}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]r = \\frac{12}{3 + 4\\cos\\theta}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"881315\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"881315\"]<\/p>\r\n\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]r = \\frac{6}{1 + \\cos\\theta}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This is already in standard form: [latex]r = \\frac{ed}{1 + e\\cos\\theta}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Identify the eccentricity by comparing with the standard form: [latex] ed = 6 \\quad \\text{and} \\quad e = 1 [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]e = 1[\/latex], this is a parabola.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]r = \\frac{8}{2 - \\sin\\theta}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">First, divide numerator and denominator by 2 to get standard form: [latex] r = \\frac{4}{1 - \\frac{1}{2}\\sin\\theta} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">This matches the form [latex]r = \\frac{ed}{1 - e\\sin\\theta}[\/latex] where: [latex] e = \\frac{1}{2} \\quad \\text{and} \\quad ed = 4 [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]e = \\frac{1}{2} &lt; 1[\/latex], this is an ellipse.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\"><\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]r = \\frac{12}{3 + 4\\cos\\theta}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Divide numerator and denominator by 3: [latex] r = \\frac{4}{1 + \\frac{4}{3}\\cos\\theta} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">This matches [latex]r = \\frac{ed}{1 + e\\cos\\theta}[\/latex] where: [latex] e = \\frac{4}{3} \\quad \\text{and} \\quad ed = 4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]e = \\frac{4}{3} &gt; 1[\/latex], this is a hyperbola.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gfebddgg-5Eh9h_8cQn4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5Eh9h_8cQn4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gfebddgg-5Eh9h_8cQn4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661476&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gfebddgg-5Eh9h_8cQn4&vembed=0&video_id=5Eh9h_8cQn4&video_target=tpm-plugin-gfebddgg-5Eh9h_8cQn4'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Identifying+a+conic+section+in+polar+coordinates_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Identifying a conic section in polar coordinates\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Graphing Polar Equations of Conics<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"63\" data-end=\"149\">Conics can be graphed directly in <strong data-start=\"97\" data-end=\"118\">polar coordinates<\/strong> using equations of the form:<\/p>\r\n\r\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi><\/mi><\/mrow><\/semantics><\/math>\r\n<p data-start=\"63\" data-end=\"149\">\\[\r\nr = \\dfrac{ed}{1 \\pm e\\cos\\theta} \\quad \\text{or} \\quad\r\nr = \\dfrac{ed}{1 \\pm e\\sin\\theta}\r\n\\]<\/p>\r\n<p data-start=\"250\" data-end=\"257\">Here:<\/p>\r\n\r\n<ul data-start=\"258\" data-end=\"607\">\r\n \t<li data-start=\"258\" data-end=\"391\">\r\n<p data-start=\"260\" data-end=\"391\">[latex]e[\/latex] = eccentricity (ellipse if [latex]e&lt;1[\/latex], parabola if [latex]e=1[\/latex], hyperbola if [latex]e&gt;1[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"392\" data-end=\"434\">\r\n<p data-start=\"394\" data-end=\"434\">[latex]d[\/latex] = directrix constant.<\/p>\r\n<\/li>\r\n \t<li data-start=\"435\" data-end=\"530\">\r\n<p data-start=\"437\" data-end=\"530\">The choice of cosine vs sine determines orientation (cosine = horizontal, sine = vertical).<\/p>\r\n<\/li>\r\n \t<li data-start=\"531\" data-end=\"607\">\r\n<p data-start=\"533\" data-end=\"607\">The sign (+\/\u2013) determines whether the conic opens right\/left or up\/down.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: How to Graph Polar Conics<\/strong>\r\n<ol>\r\n \t<li data-start=\"659\" data-end=\"828\">\r\n<p data-start=\"662\" data-end=\"695\"><strong data-start=\"662\" data-end=\"693\">Determine the Type of Conic<\/strong><\/p>\r\n\r\n<ul data-start=\"699\" data-end=\"828\">\r\n \t<li data-start=\"699\" data-end=\"728\">\r\n<p data-start=\"701\" data-end=\"728\">Compute [latex]e[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"732\" data-end=\"828\">\r\n<p data-start=\"734\" data-end=\"828\">[latex]e&lt;1[\/latex] \u2192 ellipse, [latex]e=1[\/latex] \u2192 parabola, [latex]e&gt;1[\/latex] \u2192 hyperbola.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"830\" data-end=\"1020\">\r\n<p data-start=\"833\" data-end=\"859\"><strong data-start=\"833\" data-end=\"857\">Find the Orientation<\/strong><\/p>\r\n\r\n<ul data-start=\"863\" data-end=\"1020\">\r\n \t<li data-start=\"863\" data-end=\"907\">\r\n<p data-start=\"865\" data-end=\"907\">Cosine in denominator \u2192 horizontal axis.<\/p>\r\n<\/li>\r\n \t<li data-start=\"911\" data-end=\"951\">\r\n<p data-start=\"913\" data-end=\"951\">Sine in denominator \u2192 vertical axis.<\/p>\r\n<\/li>\r\n \t<li data-start=\"955\" data-end=\"1020\">\r\n<p data-start=\"957\" data-end=\"1020\">Plus\/minus sign tells whether it opens left\/right or up\/down.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1022\" data-end=\"1186\">\r\n<p data-start=\"1025\" data-end=\"1046\"><strong data-start=\"1025\" data-end=\"1044\">Plot Key Points<\/strong><\/p>\r\n\r\n<ul data-start=\"1050\" data-end=\"1186\">\r\n \t<li data-start=\"1050\" data-end=\"1129\">\r\n<p data-start=\"1052\" data-end=\"1129\">At [latex]\\theta=0, \\pi\/2, \\pi, 3\\pi\/2[\/latex], calculate [latex]r[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1133\" data-end=\"1186\">\r\n<p data-start=\"1135\" data-end=\"1186\">Plot these points carefully in polar coordinates.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1188\" data-end=\"1369\">\r\n<p data-start=\"1191\" data-end=\"1228\"><strong data-start=\"1191\" data-end=\"1226\">Sketch the Curve Using Symmetry<\/strong><\/p>\r\n\r\n<ul data-start=\"1232\" data-end=\"1369\">\r\n \t<li data-start=\"1232\" data-end=\"1291\">\r\n<p data-start=\"1234\" data-end=\"1291\">Conics in polar form are often symmetric about an axis.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1295\" data-end=\"1369\">\r\n<p data-start=\"1297\" data-end=\"1369\">Plot enough points to capture curvature, then reflect across symmetry.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Graph the polar equation [latex]r = \\frac{4}{1 + \\cos\\theta}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"polargraph001\"]Show Solution[\/reveal-answer] [hidden-answer a=\"polargraph001\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 1: Identify the conic<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Comparing with [latex]r = \\frac{ed}{1 + e\\cos\\theta}[\/latex]: [latex]e = 1[\/latex] and [latex]ed = 4[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]e = 1[\/latex], this is a parabola.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 2: Determine orientation<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The equation uses [latex]\\cos\\theta[\/latex] with a plus sign, so the parabola opens to the left with vertex to the right of the pole.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 3: Find key points<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">At [latex]\\theta = 0[\/latex]: [latex] r = \\frac{4}{1 + \\cos 0} = \\frac{4}{1 + 1} = \\frac{4}{2} = 2 [\/latex] Point: [latex](2, 0)[\/latex] or [latex](2, 0\u00b0)[\/latex] \u2014 this is the vertex<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">At [latex]\\theta = \\frac{\\pi}{2}[\/latex]: [latex] r = \\frac{4}{1 + \\cos\\frac{\\pi}{2}} = \\frac{4}{1 + 0} = 4 [\/latex] Point: [latex](4, \\frac{\\pi}{2})[\/latex] or [latex](4, 90\u00b0)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">At [latex]\\theta = \\pi[\/latex]: [latex] r = \\frac{4}{1 + \\cos\\pi} = \\frac{4}{1 - 1} = \\frac{4}{0} = \\text{undefined} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The parabola approaches infinity as [latex]\\theta \\to \\pi[\/latex], meaning it has a vertical directrix at [latex]x = -4[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">At [latex]\\theta = \\frac{3\\pi}{2}[\/latex]: [latex] r = \\frac{4}{1 + \\cos\\frac{3\\pi}{2}} = \\frac{4}{1 + 0} = 4 [\/latex] Point: [latex](4, \\frac{3\\pi}{2})[\/latex] or [latex](4, 270\u00b0)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 4: Sketch the graph<\/p>\r\n<img class=\"alignnone wp-image-5676\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/16185203\/Screenshot-2026-02-16-at-11.51.31%E2%80%AFAM.png\" alt=\"The graph shows a parabola symmetric about the horizontal axis. The rightmost point (the vertex) occurs at (2, 0). The parabola passes through (0,4) and (0,-4)\" width=\"342\" height=\"390\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdaddbbf-zS7ZjWxu8CY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/zS7ZjWxu8CY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cdaddbbf-zS7ZjWxu8CY\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661477&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cdaddbbf-zS7ZjWxu8CY&vembed=0&video_id=zS7ZjWxu8CY&video_target=tpm-plugin-cdaddbbf-zS7ZjWxu8CY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Graphing+Conic+Sections+Using+Polar+Equations+-+Part+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Conic Sections Using Polar Equations - Part 1\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section><section class=\"textbox youChoose\" aria-label=\"You Choose\">[videopicker divId=\"tnh-video-picker\" title=\"Choose a Calculator\" label=\"Select Calculator\"]\r\n[videooption displayName=\"TI-84+\" value=\"https:\/\/www.youtube.com\/watch?v=PZwiiZQhM0c\"] [videooption displayName=\"Desmos\" value=\"https:\/\/www.youtube.com\/watch?v=rF5g4xVlo4Q\"]\r\n[\/videopicker]<\/section><\/section><\/div>\r\n<h2>Conics Defined by a Focus and Directrix<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"68\" data-end=\"466\">All conic sections\u2014parabolas, ellipses, and hyperbolas\u2014can be defined in terms of a <strong data-start=\"152\" data-end=\"161\">focus<\/strong> (a fixed point) and a <strong data-start=\"184\" data-end=\"197\">directrix<\/strong> (a fixed line). A conic is the set of all points [latex]P[\/latex] in the plane such that the ratio of the distance from [latex]P[\/latex] to the focus and the distance from [latex]P[\/latex] to the directrix is a constant [latex]e[\/latex], called the <strong data-start=\"447\" data-end=\"463\">eccentricity<\/strong>.<\/p>\r\n\r\n<ul data-start=\"468\" data-end=\"636\">\r\n \t<li data-start=\"468\" data-end=\"523\">\r\n<p data-start=\"470\" data-end=\"523\">If [latex]e=1[\/latex], the conic is a <strong data-start=\"508\" data-end=\"520\">parabola<\/strong>.<\/p>\r\n<\/li>\r\n \t<li data-start=\"524\" data-end=\"579\">\r\n<p data-start=\"526\" data-end=\"579\">If [latex]e&lt;1[\/latex], the conic is an <strong data-start=\"565\" data-end=\"576\">ellipse<\/strong>.<\/p>\r\n<\/li>\r\n \t<li data-start=\"580\" data-end=\"636\">\r\n<p data-start=\"582\" data-end=\"636\">If [latex]e&gt;1[\/latex], the conic is a <strong data-start=\"620\" data-end=\"633\">hyperbola<\/strong>.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"638\" data-end=\"708\">This definition unifies all the conics under one geometric property.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Understanding Focus-Directrix Definition<\/strong>\r\n<ol>\r\n \t<li data-start=\"775\" data-end=\"981\">\r\n<p data-start=\"778\" data-end=\"802\"><strong data-start=\"778\" data-end=\"800\">General Definition<\/strong><\/p>\r\n\r\n<ul data-start=\"806\" data-end=\"981\">\r\n \t<li data-start=\"806\" data-end=\"981\">\r\n<p data-start=\"808\" data-end=\"858\">A conic is all points [latex]P[\/latex] such that [latex]e=\\dfrac{\\text{distance from P to focus}}{\\text{distance from P to directrix}}[\/latex]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">.<\/span><\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"983\" data-end=\"1276\">\r\n<p data-start=\"986\" data-end=\"1021\"><strong data-start=\"986\" data-end=\"1019\">Special Cases by Eccentricity<\/strong><\/p>\r\n\r\n<ul data-start=\"1025\" data-end=\"1276\">\r\n \t<li data-start=\"1025\" data-end=\"1106\">\r\n<p data-start=\"1027\" data-end=\"1106\"><strong data-start=\"1027\" data-end=\"1040\">Parabola:<\/strong> Distance to focus = distance to directrix ([latex]e=1[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1110\" data-end=\"1190\">\r\n<p data-start=\"1112\" data-end=\"1190\"><strong data-start=\"1112\" data-end=\"1124\">Ellipse:<\/strong> Distance to focus &lt; distance to directrix ([latex]e&lt;1[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1194\" data-end=\"1276\">\r\n<p data-start=\"1196\" data-end=\"1276\"><strong data-start=\"1196\" data-end=\"1210\">Hyperbola:<\/strong> Distance to focus &gt; distance to directrix ([latex]e&gt;1[\/latex]).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A conic has its focus at the origin and directrix [latex]x = 4[\/latex]. A point on the conic is twice as far from the directrix as it is from the focus. Find the eccentricity and identify the conic.<\/p>\r\n[reveal-answer q=\"42469\"]Show Solution[\/reveal-answer][hidden-answer a=\"42469\"]\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">By definition, for any point [latex]P[\/latex] on the conic: [latex] e = \\frac{\\text{distance from P to focus}}{\\text{distance from P to directrix}} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">We're told the point is twice as far from the directrix as from the focus: [latex] \\text{distance to directrix} = 2 \\times \\text{distance to focus} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Rearranging: [latex] \\frac{\\text{distance to focus}}{\\text{distance to directrix}} = \\frac{1}{2} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Therefore: [latex] e = \\frac{1}{2} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]e = \\frac{1}{2} &lt; 1[\/latex], this is an ellipse.<\/p>\r\n[\/hidden-answer]\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<h2>Identifying a Conic in Polar Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"62\" data-end=\"208\">Conics can also be expressed in <strong data-start=\"94\" data-end=\"108\">polar form<\/strong>, where the focus is placed at the pole (origin). In this setting, the general form of a conic is:<\/p>\n<p data-start=\"62\" data-end=\"208\">\\[<br \/>\nr = \\dfrac{ed}{1 \\pm e\\cos\\theta} \\quad \\text{or} \\quad<br \/>\nr = \\dfrac{ed}{1 \\pm e\\sin\\theta}<br \/>\n\\]<\/p>\n<p data-start=\"309\" data-end=\"316\">Here:<\/p>\n<ul data-start=\"317\" data-end=\"567\">\n<li data-start=\"317\" data-end=\"384\">\n<p data-start=\"319\" data-end=\"384\">[latex]e[\/latex] = eccentricity (determines the type of conic).<\/p>\n<\/li>\n<li data-start=\"385\" data-end=\"466\">\n<p data-start=\"387\" data-end=\"466\">[latex]d[\/latex] = directrix constant (distance related to conic definition).<\/p>\n<\/li>\n<li data-start=\"467\" data-end=\"567\">\n<p data-start=\"469\" data-end=\"567\">The sign\/choice of cosine or sine determines orientation: right\/left (cosine) or up\/down (sine).<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: How to Identify the Conic<\/strong><\/p>\n<ol>\n<li data-start=\"619\" data-end=\"805\">\n<p data-start=\"622\" data-end=\"652\"><strong data-start=\"622\" data-end=\"650\">Look at the Eccentricity<\/strong><\/p>\n<ul data-start=\"656\" data-end=\"805\">\n<li data-start=\"656\" data-end=\"690\">\n<p data-start=\"658\" data-end=\"690\">[latex]e=1[\/latex] \u2192 parabola.<\/p>\n<\/li>\n<li data-start=\"694\" data-end=\"766\">\n<p data-start=\"696\" data-end=\"766\">[latex]e<1[\/latex] \u2192 ellipse (if [latex]e=0[\/latex], it\u2019s a circle).<\/p>\n<\/li>\n<li data-start=\"770\" data-end=\"805\">\n<p data-start=\"772\" data-end=\"805\">[latex]e>1[\/latex] \u2192 hyperbola.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"807\" data-end=\"1077\">\n<p data-start=\"810\" data-end=\"833\"><strong data-start=\"810\" data-end=\"831\">Check Orientation<\/strong><\/p>\n<ul data-start=\"837\" data-end=\"1077\">\n<li data-start=\"837\" data-end=\"922\">\n<p data-start=\"839\" data-end=\"922\">[latex]r=\\dfrac{ed}{1+e\\cos\\theta}[\/latex] \u2192 focus at origin, directrix vertical.<\/p>\n<\/li>\n<li data-start=\"926\" data-end=\"996\">\n<p data-start=\"928\" data-end=\"996\">[latex]r=\\dfrac{ed}{1-e\\cos\\theta}[\/latex] \u2192 similar but mirrored.<\/p>\n<\/li>\n<li data-start=\"1000\" data-end=\"1077\">\n<p data-start=\"1002\" data-end=\"1077\">[latex]r=\\dfrac{ed}{1\\pm e\\sin\\theta}[\/latex] \u2192 opens upward or downward.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1707\" data-end=\"1869\">\n<p data-start=\"1710\" data-end=\"1728\"><strong data-start=\"1710\" data-end=\"1726\">Summary Rule<\/strong><\/p>\n<ul data-start=\"1732\" data-end=\"1869\">\n<li data-start=\"1732\" data-end=\"1798\">\n<p data-start=\"1734\" data-end=\"1798\">The <em data-start=\"1738\" data-end=\"1752\">eccentricity<\/em> [latex]e[\/latex] tells you <strong data-start=\"1780\" data-end=\"1795\">which conic<\/strong>.<\/p>\n<\/li>\n<li data-start=\"1802\" data-end=\"1869\">\n<p data-start=\"1804\" data-end=\"1869\">The <em data-start=\"1808\" data-end=\"1837\">function (cos\/sin) and sign<\/em> tell you <strong data-start=\"1847\" data-end=\"1866\">which direction<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<div>\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Identify the conic represented by each polar equation.<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]r = \\frac{6}{1 + \\cos\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]r = \\frac{8}{2 - \\sin\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]r = \\frac{12}{3 + 4\\cos\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q881315\">Show Solution<\/button><\/p>\n<div id=\"q881315\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">a) [latex]r = \\frac{6}{1 + \\cos\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">This is already in standard form: [latex]r = \\frac{ed}{1 + e\\cos\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Identify the eccentricity by comparing with the standard form: [latex]ed = 6 \\quad \\text{and} \\quad e = 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]e = 1[\/latex], this is a parabola.<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">b) [latex]r = \\frac{8}{2 - \\sin\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">First, divide numerator and denominator by 2 to get standard form: [latex]r = \\frac{4}{1 - \\frac{1}{2}\\sin\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">This matches the form [latex]r = \\frac{ed}{1 - e\\sin\\theta}[\/latex] where: [latex]e = \\frac{1}{2} \\quad \\text{and} \\quad ed = 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]e = \\frac{1}{2} < 1[\/latex], this is an ellipse.<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\"><\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">c) [latex]r = \\frac{12}{3 + 4\\cos\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Divide numerator and denominator by 3: [latex]r = \\frac{4}{1 + \\frac{4}{3}\\cos\\theta}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">This matches [latex]r = \\frac{ed}{1 + e\\cos\\theta}[\/latex] where: [latex]e = \\frac{4}{3} \\quad \\text{and} \\quad ed = 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid &#091;&amp;_&gt;_*&#093;:min-w-0 gap-3\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]e = \\frac{4}{3} > 1[\/latex], this is a hyperbola.<\/p>\n<\/div>\n<\/div>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gfebddgg-5Eh9h_8cQn4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5Eh9h_8cQn4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gfebddgg-5Eh9h_8cQn4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661476&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gfebddgg-5Eh9h_8cQn4&#38;vembed=0&#38;video_id=5Eh9h_8cQn4&#38;video_target=tpm-plugin-gfebddgg-5Eh9h_8cQn4\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Identifying+a+conic+section+in+polar+coordinates_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Identifying a conic section in polar coordinates\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Graphing Polar Equations of Conics<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"63\" data-end=\"149\">Conics can be graphed directly in <strong data-start=\"97\" data-end=\"118\">polar coordinates<\/strong> using equations of the form:<\/p>\n<p><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi><\/mi><\/mrow><\/semantics><\/math><\/p>\n<p data-start=\"63\" data-end=\"149\">\\[<br \/>\nr = \\dfrac{ed}{1 \\pm e\\cos\\theta} \\quad \\text{or} \\quad<br \/>\nr = \\dfrac{ed}{1 \\pm e\\sin\\theta}<br \/>\n\\]<\/p>\n<p data-start=\"250\" data-end=\"257\">Here:<\/p>\n<ul data-start=\"258\" data-end=\"607\">\n<li data-start=\"258\" data-end=\"391\">\n<p data-start=\"260\" data-end=\"391\">[latex]e[\/latex] = eccentricity (ellipse if [latex]e<1[\/latex], parabola if [latex]e=1[\/latex], hyperbola if [latex]e>1[\/latex]).<\/p>\n<\/li>\n<li data-start=\"392\" data-end=\"434\">\n<p data-start=\"394\" data-end=\"434\">[latex]d[\/latex] = directrix constant.<\/p>\n<\/li>\n<li data-start=\"435\" data-end=\"530\">\n<p data-start=\"437\" data-end=\"530\">The choice of cosine vs sine determines orientation (cosine = horizontal, sine = vertical).<\/p>\n<\/li>\n<li data-start=\"531\" data-end=\"607\">\n<p data-start=\"533\" data-end=\"607\">The sign (+\/\u2013) determines whether the conic opens right\/left or up\/down.<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: How to Graph Polar Conics<\/strong><\/p>\n<ol>\n<li data-start=\"659\" data-end=\"828\">\n<p data-start=\"662\" data-end=\"695\"><strong data-start=\"662\" data-end=\"693\">Determine the Type of Conic<\/strong><\/p>\n<ul data-start=\"699\" data-end=\"828\">\n<li data-start=\"699\" data-end=\"728\">\n<p data-start=\"701\" data-end=\"728\">Compute [latex]e[\/latex].<\/p>\n<\/li>\n<li data-start=\"732\" data-end=\"828\">\n<p data-start=\"734\" data-end=\"828\">[latex]e<1[\/latex] \u2192 ellipse, [latex]e=1[\/latex] \u2192 parabola, [latex]e>1[\/latex] \u2192 hyperbola.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"830\" data-end=\"1020\">\n<p data-start=\"833\" data-end=\"859\"><strong data-start=\"833\" data-end=\"857\">Find the Orientation<\/strong><\/p>\n<ul data-start=\"863\" data-end=\"1020\">\n<li data-start=\"863\" data-end=\"907\">\n<p data-start=\"865\" data-end=\"907\">Cosine in denominator \u2192 horizontal axis.<\/p>\n<\/li>\n<li data-start=\"911\" data-end=\"951\">\n<p data-start=\"913\" data-end=\"951\">Sine in denominator \u2192 vertical axis.<\/p>\n<\/li>\n<li data-start=\"955\" data-end=\"1020\">\n<p data-start=\"957\" data-end=\"1020\">Plus\/minus sign tells whether it opens left\/right or up\/down.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1022\" data-end=\"1186\">\n<p data-start=\"1025\" data-end=\"1046\"><strong data-start=\"1025\" data-end=\"1044\">Plot Key Points<\/strong><\/p>\n<ul data-start=\"1050\" data-end=\"1186\">\n<li data-start=\"1050\" data-end=\"1129\">\n<p data-start=\"1052\" data-end=\"1129\">At [latex]\\theta=0, \\pi\/2, \\pi, 3\\pi\/2[\/latex], calculate [latex]r[\/latex].<\/p>\n<\/li>\n<li data-start=\"1133\" data-end=\"1186\">\n<p data-start=\"1135\" data-end=\"1186\">Plot these points carefully in polar coordinates.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1188\" data-end=\"1369\">\n<p data-start=\"1191\" data-end=\"1228\"><strong data-start=\"1191\" data-end=\"1226\">Sketch the Curve Using Symmetry<\/strong><\/p>\n<ul data-start=\"1232\" data-end=\"1369\">\n<li data-start=\"1232\" data-end=\"1291\">\n<p data-start=\"1234\" data-end=\"1291\">Conics in polar form are often symmetric about an axis.<\/p>\n<\/li>\n<li data-start=\"1295\" data-end=\"1369\">\n<p data-start=\"1297\" data-end=\"1369\">Plot enough points to capture curvature, then reflect across symmetry.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Graph the polar equation [latex]r = \\frac{4}{1 + \\cos\\theta}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qpolargraph001\">Show Solution<\/button> <\/p>\n<div id=\"qpolargraph001\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 1: Identify the conic<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Comparing with [latex]r = \\frac{ed}{1 + e\\cos\\theta}[\/latex]: [latex]e = 1[\/latex] and [latex]ed = 4[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]e = 1[\/latex], this is a parabola.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 2: Determine orientation<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The equation uses [latex]\\cos\\theta[\/latex] with a plus sign, so the parabola opens to the left with vertex to the right of the pole.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 3: Find key points<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">At [latex]\\theta = 0[\/latex]: [latex]r = \\frac{4}{1 + \\cos 0} = \\frac{4}{1 + 1} = \\frac{4}{2} = 2[\/latex] Point: [latex](2, 0)[\/latex] or [latex](2, 0\u00b0)[\/latex] \u2014 this is the vertex<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">At [latex]\\theta = \\frac{\\pi}{2}[\/latex]: [latex]r = \\frac{4}{1 + \\cos\\frac{\\pi}{2}} = \\frac{4}{1 + 0} = 4[\/latex] Point: [latex](4, \\frac{\\pi}{2})[\/latex] or [latex](4, 90\u00b0)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">At [latex]\\theta = \\pi[\/latex]: [latex]r = \\frac{4}{1 + \\cos\\pi} = \\frac{4}{1 - 1} = \\frac{4}{0} = \\text{undefined}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The parabola approaches infinity as [latex]\\theta \\to \\pi[\/latex], meaning it has a vertical directrix at [latex]x = -4[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">At [latex]\\theta = \\frac{3\\pi}{2}[\/latex]: [latex]r = \\frac{4}{1 + \\cos\\frac{3\\pi}{2}} = \\frac{4}{1 + 0} = 4[\/latex] Point: [latex](4, \\frac{3\\pi}{2})[\/latex] or [latex](4, 270\u00b0)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 4: Sketch the graph<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5676\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/16185203\/Screenshot-2026-02-16-at-11.51.31%E2%80%AFAM.png\" alt=\"The graph shows a parabola symmetric about the horizontal axis. The rightmost point (the vertex) occurs at (2, 0). The parabola passes through (0,4) and (0,-4)\" width=\"342\" height=\"390\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/16185203\/Screenshot-2026-02-16-at-11.51.31%E2%80%AFAM.png 638w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/16185203\/Screenshot-2026-02-16-at-11.51.31%E2%80%AFAM-263x300.png 263w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/16185203\/Screenshot-2026-02-16-at-11.51.31%E2%80%AFAM-65x74.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/16185203\/Screenshot-2026-02-16-at-11.51.31%E2%80%AFAM-225x257.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/16185203\/Screenshot-2026-02-16-at-11.51.31%E2%80%AFAM-350x399.png 350w\" sizes=\"(max-width: 342px) 100vw, 342px\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdaddbbf-zS7ZjWxu8CY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/zS7ZjWxu8CY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cdaddbbf-zS7ZjWxu8CY\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661477&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cdaddbbf-zS7ZjWxu8CY&#38;vembed=0&#38;video_id=zS7ZjWxu8CY&#38;video_target=tpm-plugin-cdaddbbf-zS7ZjWxu8CY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Graphing+Conic+Sections+Using+Polar+Equations+-+Part+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Conic Sections Using Polar Equations &#8211; Part 1\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section>\n<section class=\"textbox youChoose\" aria-label=\"You Choose\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Choose a Calculator<\/h3>\n<form><label>Select Calculator:<\/label><select name=\"video\"><option value=\"https:\/\/www.youtube.com\/embed\/PZwiiZQhM0c\">TI-84+<\/option><option value=\"https:\/\/www.youtube.com\/embed\/rF5g4xVlo4Q\">Desmos<\/option><\/select><\/form>\n<div class=\"videoContainer\"><iframe src=\"https:\/\/www.youtube.com\/embed\/PZwiiZQhM0c\" allowfullscreen><\/iframe><\/div>\n<\/section>\n<\/section>\n<\/div>\n<h2>Conics Defined by a Focus and Directrix<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"68\" data-end=\"466\">All conic sections\u2014parabolas, ellipses, and hyperbolas\u2014can be defined in terms of a <strong data-start=\"152\" data-end=\"161\">focus<\/strong> (a fixed point) and a <strong data-start=\"184\" data-end=\"197\">directrix<\/strong> (a fixed line). A conic is the set of all points [latex]P[\/latex] in the plane such that the ratio of the distance from [latex]P[\/latex] to the focus and the distance from [latex]P[\/latex] to the directrix is a constant [latex]e[\/latex], called the <strong data-start=\"447\" data-end=\"463\">eccentricity<\/strong>.<\/p>\n<ul data-start=\"468\" data-end=\"636\">\n<li data-start=\"468\" data-end=\"523\">\n<p data-start=\"470\" data-end=\"523\">If [latex]e=1[\/latex], the conic is a <strong data-start=\"508\" data-end=\"520\">parabola<\/strong>.<\/p>\n<\/li>\n<li data-start=\"524\" data-end=\"579\">\n<p data-start=\"526\" data-end=\"579\">If [latex]e<1[\/latex], the conic is an <strong data-start=\"565\" data-end=\"576\">ellipse<\/strong>.<\/p>\n<\/li>\n<li data-start=\"580\" data-end=\"636\">\n<p data-start=\"582\" data-end=\"636\">If [latex]e>1[\/latex], the conic is a <strong data-start=\"620\" data-end=\"633\">hyperbola<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"638\" data-end=\"708\">This definition unifies all the conics under one geometric property.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Understanding Focus-Directrix Definition<\/strong><\/p>\n<ol>\n<li data-start=\"775\" data-end=\"981\">\n<p data-start=\"778\" data-end=\"802\"><strong data-start=\"778\" data-end=\"800\">General Definition<\/strong><\/p>\n<ul data-start=\"806\" data-end=\"981\">\n<li data-start=\"806\" data-end=\"981\">\n<p data-start=\"808\" data-end=\"858\">A conic is all points [latex]P[\/latex] such that [latex]e=\\dfrac{\\text{distance from P to focus}}{\\text{distance from P to directrix}}[\/latex]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"983\" data-end=\"1276\">\n<p data-start=\"986\" data-end=\"1021\"><strong data-start=\"986\" data-end=\"1019\">Special Cases by Eccentricity<\/strong><\/p>\n<ul data-start=\"1025\" data-end=\"1276\">\n<li data-start=\"1025\" data-end=\"1106\">\n<p data-start=\"1027\" data-end=\"1106\"><strong data-start=\"1027\" data-end=\"1040\">Parabola:<\/strong> Distance to focus = distance to directrix ([latex]e=1[\/latex]).<\/p>\n<\/li>\n<li data-start=\"1110\" data-end=\"1190\">\n<p data-start=\"1112\" data-end=\"1190\"><strong data-start=\"1112\" data-end=\"1124\">Ellipse:<\/strong> Distance to focus &lt; distance to directrix ([latex]e<1[\/latex]).<\/p>\n<\/li>\n<li data-start=\"1194\" data-end=\"1276\">\n<p data-start=\"1196\" data-end=\"1276\"><strong data-start=\"1196\" data-end=\"1210\">Hyperbola:<\/strong> Distance to focus &gt; distance to directrix ([latex]e>1[\/latex]).<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A conic has its focus at the origin and directrix [latex]x = 4[\/latex]. A point on the conic is twice as far from the directrix as it is from the focus. Find the eccentricity and identify the conic.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q42469\">Show Solution<\/button><\/p>\n<div id=\"q42469\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">By definition, for any point [latex]P[\/latex] on the conic: [latex]e = \\frac{\\text{distance from P to focus}}{\\text{distance from P to directrix}}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">We&#8217;re told the point is twice as far from the directrix as from the focus: [latex]\\text{distance to directrix} = 2 \\times \\text{distance to focus}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Rearranging: [latex]\\frac{\\text{distance to focus}}{\\text{distance to directrix}} = \\frac{1}{2}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Therefore: [latex]e = \\frac{1}{2}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]e = \\frac{1}{2} < 1[\/latex], this is an ellipse.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":32,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Example: Identifying a conic section in polar coordinates\",\"author\":\"Justin Ryan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/5Eh9h_8cQn4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Graphing Conic Sections Using Polar Equations - 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