{"id":1045,"date":"2025-06-20T17:30:28","date_gmt":"2025-06-20T17:30:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1045"},"modified":"2025-09-09T19:40:54","modified_gmt":"2025-09-09T19:40:54","slug":"conic-sections-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/conic-sections-fresh-take\/","title":{"raw":"Conic Sections: Fresh Take","rendered":"Conic Sections: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write the equation of a parabola when you know its focus and directrix<\/li>\r\n \t<li>Write the equation of an ellipse when you know its foci<\/li>\r\n \t<li>Write the equation of a hyperbola when you know its foci<\/li>\r\n \t<li>Identify which type of conic section you have based on its eccentricity value<\/li>\r\n \t<li>Write polar equations for conic sections using eccentricity<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Parabolas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A parabola isn't just a U-shaped curve\u2014it's defined by a beautiful geometric relationship. Every point on a parabola sits exactly the same distance from two things: a fixed point called the <strong>focus<\/strong> and a fixed line called the <strong>directrix<\/strong>.<\/p>\r\n<p class=\"whitespace-normal break-words\">For any point [latex]P[\/latex] on the parabola, the distance from [latex]P[\/latex] to the focus equals the distance from [latex]P[\/latex] to the directrix. This constant-distance relationship creates the parabola's characteristic shape.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Standard form equations:<\/strong> Once you know the vertex [latex](h,k)[\/latex] and the focus distance [latex]p[\/latex]:<\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Opens up\/down:<\/strong> [latex]y = \\frac{1}{4p}(x-h)^2 + k[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Opens left\/right:<\/strong> [latex]x = \\frac{1}{4p}(y-k)^2 + h[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Key relationships:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vertex:<\/strong> Halfway between focus and directrix at [latex](h,k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Focus:<\/strong> Distance [latex]p[\/latex] from vertex toward the parabola's opening<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Directrix:<\/strong> Line distance [latex]p[\/latex] from vertex away from the parabola's opening<\/li>\r\n<\/ul>\r\nThis focus-directrix relationship creates parabolas' most useful feature\u2014parallel rays entering the parabola all reflect to the focus. This is why satellite dishes, car headlights, and solar collectors all use parabolic shapes.\r\n<p class=\"whitespace-normal break-words\"><strong>Converting from general form:<\/strong> Use completing the square to transform equations like [latex]x^2 - 4x - 8y + 12 = 0[\/latex] into standard form. Group the squared variable terms, complete the square, then solve for the other variable.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1167794023921\" data-type=\"problem\">\r\n<p id=\"fs-id1167794332814\">Put the equation [latex]2{y}^{2}-x+12y+16=0[\/latex] into standard form and graph the resulting parabola.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558897\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1167793366326\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167793966387\">Solve for <em data-effect=\"italics\">x<\/em>. Check which direction the parabola opens.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1167793976664\" data-type=\"solution\">\r\n<p id=\"fs-id1167794046069\" style=\"text-align: center;\">[latex]x=2{\\left(y+3\\right)}^{2}-2[\/latex] <span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"342\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225338\/CNX_Calc_Figure_11_05_006.jpg\" alt=\"A parabola is drawn with vertex at (\u22122, \u22123) and opening to the right with equation x = 2(y + 3)2 \u2013 2. The focus is drawn at (0, \u22123). The directrix is drawn at x = \u22124.\" width=\"342\" height=\"236\" data-media-type=\"image\/jpeg\" \/> Figure 6.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Ellipses<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">An ellipse has a beautifully simple defining property: every point on the ellipse maintains a constant sum of distances to two fixed points called foci. This constant-sum relationship creates the ellipse's characteristic oval shape.<\/p>\r\n<p class=\"whitespace-normal break-words\">For any point [latex]P[\/latex] on the ellipse, [latex]d(P, F_1) + d(P, F_2) = 2a[\/latex], where [latex]a[\/latex] is the semi-major axis length. This sum never changes, no matter which point you choose on the ellipse.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Key measurements:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Major axis:<\/strong> The longest diameter, with length [latex]2a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Minor axis:<\/strong> The shortest diameter, with length [latex]2b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Distance between foci:<\/strong> [latex]2c[\/latex], where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\r\n<\/ul>\r\nAlways remember [latex]c^2 = a^2 - b^2[\/latex]. This connects the focus distance to the axis lengths and ensures [latex]c &lt; a[\/latex] (foci are always inside the ellipse).\r\n<p class=\"whitespace-normal break-words\"><strong>Standard form equations:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Horizontal major axis:<\/strong> [latex]\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vertical major axis:<\/strong> [latex]\\frac{(x-h)^2}{b^2} + \\frac{(y-k)^2}{a^2} = 1[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">In standard form, the larger denominator tells you the major axis direction. If [latex]a^2[\/latex] is under the [latex]x[\/latex]-term, the major axis is horizontal; if under the [latex]y[\/latex]-term, it's vertical.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Converting from general form:<\/strong> Use completing the square on both [latex]x[\/latex] and [latex]y[\/latex] terms separately. Group like terms, factor out coefficients, complete the square for each variable, then divide to get the equation equal to 1.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1167793373866\" data-type=\"problem\">\r\n<p id=\"fs-id1167793373868\">Put the equation [latex]9{x}^{2}+16{y}^{2}+18x - 64y - 71=0[\/latex] into standard form and graph the resulting ellipse.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558894\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1167793240741\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167793287354\">Move the constant over and complete the square.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1167793263801\" data-type=\"solution\">\r\n<p id=\"fs-id1167793263804\" style=\"text-align: center;\">[latex]\\frac{{\\left(x+1\\right)}^{2}}{16}+\\frac{{\\left(y - 2\\right)}^{2}}{9}=1[\/latex] <span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"430\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225348\/CNX_Calc_Figure_11_05_010.jpg\" alt=\"An ellipse is drawn with equation 9x2 + 16y2 + 18x \u2013 64y \u2212 71 = 0. It has center at (\u22121, 2), touches the x-axis at (2, 0) and (\u22124, 0), and touches the y-axis near (0, \u22121) and (0, 5).\" width=\"430\" height=\"347\" data-media-type=\"image\/jpeg\" \/> Figure 10.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Hyperbolas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A hyperbola is defined by a difference relationship rather than a sum. Every point on a hyperbola maintains a constant difference between its distances to two fixed points called foci. This difference-based definition creates the hyperbola's distinctive two-branch shape.<\/p>\r\n<p class=\"whitespace-normal break-words\">For any point [latex]P[\/latex] on the hyperbola, [latex]|d(P, F_1) - d(P, F_2)| = 2a[\/latex], where [latex]a[\/latex] is the distance from center to vertex. This difference never changes, no matter which point you choose.<\/p>\r\nUnlike ellipses (which are \"closed\" curves), hyperbolas have two separate branches that extend infinitely, approaching but never touching their asymptotes.\r\n<p class=\"whitespace-normal break-words\"><strong>Key relationship:<\/strong> For hyperbolas, [latex]c^2 = a^2 + b^2[\/latex] (note the plus sign, unlike ellipses). This means [latex]c &gt; a[\/latex] always\u2014the foci are farther from the center than the vertices are.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Standard form equations:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Horizontal transverse axis:<\/strong> [latex]\\frac{(x-h)^2}{a^2} - \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vertical transverse axis:<\/strong> [latex]\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The positive term tells you the transverse axis direction. If [latex]x[\/latex] is positive, the hyperbola opens left-right; if [latex]y[\/latex] is positive, it opens up-down.<\/p>\r\n<p class=\"whitespace-normal break-words\">Asymptotes are crucial.\u00a0They guide the hyperbola's shape. For horizontal hyperbolas: [latex]y = k \\pm \\frac{b}{a}(x-h)[\/latex]. For vertical hyperbolas: [latex]y = k \\pm \\frac{a}{b}(x-h)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Converting from general form:<\/strong> Use completing the square on both variables, but watch the signs carefully. One squared term is positive, one is negative.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1167793979474\" data-type=\"problem\">\r\n<p id=\"fs-id1167793979476\">Put the equation [latex]4{y}^{2}-9{x}^{2}+16y+18x - 29=0[\/latex] into standard form and graph the resulting hyperbola. What are the equations of the asymptotes?<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558891\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558891\"]\r\n<div id=\"fs-id1167794159385\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167794159392\">Move the constant over and complete the square. Check which direction the hyperbola opens.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558892\"]\r\n<div id=\"fs-id1167793979522\" data-type=\"solution\">\r\n<p id=\"fs-id1167793979524\" style=\"text-align: center;\">[latex]\\frac{{\\left(y+2\\right)}^{2}}{9}-\\frac{{\\left(x - 1\\right)}^{2}}{4}=1[\/latex]. This is a vertical hyperbola. Asymptotes [latex]y=-2\\pm \\frac{3}{2}\\left(x - 1\\right)[\/latex]. <span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"454\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225359\/CNX_Calc_Figure_11_05_014.jpg\" alt=\"A hyperbola is drawn with equation 4y2 \u2013 9x2 + 16x + 18y \u2013 29 = 0. It has center at (1, \u22122), and the hyperbolas are open to the top and bottom.\" width=\"454\" height=\"534\" data-media-type=\"image\/jpeg\" \/> Figure 14.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Eccentricity and Directrix<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Eccentricity provides a single number that tells you exactly which type of conic section you're dealing with. It's defined as the constant ratio of distance-to-focus over distance-to-directrix for any point on the conic.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The eccentricity formula:<\/strong> [latex]e = \\frac{\\text{distance from point to focus}}{\\text{distance from point to directrix}}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The classification system:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]e = 0[\/latex]: Circle (special case of ellipse)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0 &lt; e &lt; 1[\/latex]: Ellipse<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]e = 1[\/latex]: Parabola<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]e &gt; 1[\/latex]: Hyperbola<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Computing eccentricity from standard form:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Ellipses and circles:<\/strong> [latex]e = \\frac{c}{a}[\/latex] where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Hyperbolas:<\/strong> [latex]e = \\frac{c}{a}[\/latex] where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parabolas:<\/strong> [latex]e = 1[\/latex] always<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Directrix locations:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parabolas:<\/strong> One directrix at distance [latex]p[\/latex] from vertex, opposite the focus<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Ellipses\/Hyperbolas:<\/strong> Two directrices at [latex]x = h \\pm \\frac{a^2}{c}[\/latex] (horizontal) or [latex]y = k \\pm \\frac{a^2}{c}[\/latex] (vertical)<\/li>\r\n<\/ul>\r\nEccentricity measures how \"stretched\" a conic is. A circle ([latex]e = 0[\/latex]) has no stretch. As [latex]e[\/latex] approaches 1, an ellipse becomes more elongated. A parabola ([latex]e = 1[\/latex]) represents the boundary case. Hyperbolas ([latex]e &gt; 1[\/latex]) are \"stretched beyond the breaking point\" into separate branches. Think of eccentricity as measuring how far the conic deviates from being a perfect circle. The closer to 0, the more circular; the farther from 0, the more \"eccentric\" the shape becomes.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1167794138959\" data-type=\"problem\">\r\n<p id=\"fs-id1167794138961\">Determine the eccentricity of the hyperbola described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167794138965\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{{\\left(y - 3\\right)}^{2}}{49}-\\frac{{\\left(x+2\\right)}^{2}}{25}=1[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558879\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558879\"]\r\n<div id=\"fs-id1167794139057\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167794069860\">First find the values of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, then determine <em data-effect=\"italics\">c<\/em> using the equation [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558889\"]\r\n<div id=\"fs-id1167794139025\" data-type=\"solution\">\r\n<p id=\"fs-id1167794139028\" style=\"text-align: center;\">[latex]e=\\frac{c}{a}=\\frac{\\sqrt{74}}{7}\\approx 1.229[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Polar Equations of Conic Sections<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">All conic sections can be written in a single, elegant polar form that directly incorporates their defining geometric properties. This unified approach reveals the deep connections between parabolas, ellipses, and hyperbolas.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The universal polar equation:<\/strong> [latex]r = \\frac{ep}{1 \\pm e\\cos\\theta}[\/latex] or [latex]r = \\frac{ep}{1 \\pm e\\sin\\theta}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">where [latex]e[\/latex] is the eccentricity and [latex]p[\/latex] is the focal parameter (distance from focus to directrix).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Analyzing the equation:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Normalize the denominator:<\/strong> Make the constant term equal to 1 by factoring<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify eccentricity:<\/strong> The coefficient of the trig function is [latex]e[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Determine orientation:<\/strong> Cosine means horizontal major axis; sine means vertical major axis<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>What each part tells you:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]e = 0[\/latex]: Circle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0 &lt; e &lt; 1[\/latex]: Ellipse<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]e = 1[\/latex]: Parabola<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]e &gt; 1[\/latex]: Hyperbola<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>The focal parameter [latex]p[\/latex]:<\/strong><\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parabolas:<\/strong> [latex]p = 2a[\/latex] (twice the distance from vertex to focus)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Ellipses:<\/strong> [latex]p = \\frac{a(1-e^2)}{e}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Hyperbolas:<\/strong> [latex]p = \\frac{a(e^2-1)}{e}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1167793380150\" data-type=\"problem\">\r\n<p id=\"fs-id1167793380152\">Identify and create a graph of the conic section described by the equation<\/p>\r\n\r\n<div id=\"fs-id1167793380155\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\frac{4}{1 - 0.8\\sin\\theta }[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558849\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558849\"]\r\n<div id=\"fs-id1167794172253\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1167794172261\">First find the values of <em data-effect=\"italics\">e<\/em> and <em data-effect=\"italics\">p<\/em>, and then create a table of values.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558859\"]\r\n<div id=\"fs-id1167793380185\" data-type=\"solution\">\r\n<p id=\"fs-id1167793380187\">Here [latex]e=0.8[\/latex] and [latex]p=5[\/latex]. This conic section is an ellipse.<span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"424\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225411\/CNX_Calc_Figure_11_05_018.jpg\" alt=\"Graph of an ellipse with equation r = 4\/(1 \u2013 0.8 sin\u03b8), center near (0, 11), major axis roughly 22, and minor axis roughly 12.\" width=\"424\" height=\"572\" data-media-type=\"image\/jpeg\" \/> Figure 18.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write the equation of a parabola when you know its focus and directrix<\/li>\n<li>Write the equation of an ellipse when you know its foci<\/li>\n<li>Write the equation of a hyperbola when you know its foci<\/li>\n<li>Identify which type of conic section you have based on its eccentricity value<\/li>\n<li>Write polar equations for conic sections using eccentricity<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Parabolas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A parabola isn&#8217;t just a U-shaped curve\u2014it&#8217;s defined by a beautiful geometric relationship. Every point on a parabola sits exactly the same distance from two things: a fixed point called the <strong>focus<\/strong> and a fixed line called the <strong>directrix<\/strong>.<\/p>\n<p class=\"whitespace-normal break-words\">For any point [latex]P[\/latex] on the parabola, the distance from [latex]P[\/latex] to the focus equals the distance from [latex]P[\/latex] to the directrix. This constant-distance relationship creates the parabola&#8217;s characteristic shape.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Standard form equations:<\/strong> Once you know the vertex [latex](h,k)[\/latex] and the focus distance [latex]p[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Opens up\/down:<\/strong> [latex]y = \\frac{1}{4p}(x-h)^2 + k[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Opens left\/right:<\/strong> [latex]x = \\frac{1}{4p}(y-k)^2 + h[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Key relationships:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Vertex:<\/strong> Halfway between focus and directrix at [latex](h,k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Focus:<\/strong> Distance [latex]p[\/latex] from vertex toward the parabola&#8217;s opening<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Directrix:<\/strong> Line distance [latex]p[\/latex] from vertex away from the parabola&#8217;s opening<\/li>\n<\/ul>\n<p>This focus-directrix relationship creates parabolas&#8217; most useful feature\u2014parallel rays entering the parabola all reflect to the focus. This is why satellite dishes, car headlights, and solar collectors all use parabolic shapes.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Converting from general form:<\/strong> Use completing the square to transform equations like [latex]x^2 - 4x - 8y + 12 = 0[\/latex] into standard form. Group the squared variable terms, complete the square, then solve for the other variable.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1167794023921\" data-type=\"problem\">\n<p id=\"fs-id1167794332814\">Put the equation [latex]2{y}^{2}-x+12y+16=0[\/latex] into standard form and graph the resulting parabola.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Hint<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793366326\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167793966387\">Solve for <em data-effect=\"italics\">x<\/em>. Check which direction the parabola opens.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Show Solution<\/button><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793976664\" data-type=\"solution\">\n<p id=\"fs-id1167794046069\" style=\"text-align: center;\">[latex]x=2{\\left(y+3\\right)}^{2}-2[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 342px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225338\/CNX_Calc_Figure_11_05_006.jpg\" alt=\"A parabola is drawn with vertex at (\u22122, \u22123) and opening to the right with equation x = 2(y + 3)2 \u2013 2. The focus is drawn at (0, \u22123). The directrix is drawn at x = \u22124.\" width=\"342\" height=\"236\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 6.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Ellipses<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">An ellipse has a beautifully simple defining property: every point on the ellipse maintains a constant sum of distances to two fixed points called foci. This constant-sum relationship creates the ellipse&#8217;s characteristic oval shape.<\/p>\n<p class=\"whitespace-normal break-words\">For any point [latex]P[\/latex] on the ellipse, [latex]d(P, F_1) + d(P, F_2) = 2a[\/latex], where [latex]a[\/latex] is the semi-major axis length. This sum never changes, no matter which point you choose on the ellipse.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Key measurements:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Major axis:<\/strong> The longest diameter, with length [latex]2a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Minor axis:<\/strong> The shortest diameter, with length [latex]2b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Distance between foci:<\/strong> [latex]2c[\/latex], where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\n<\/ul>\n<p>Always remember [latex]c^2 = a^2 - b^2[\/latex]. This connects the focus distance to the axis lengths and ensures [latex]c < a[\/latex] (foci are always inside the ellipse).\n\n\n<p class=\"whitespace-normal break-words\"><strong>Standard form equations:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Horizontal major axis:<\/strong> [latex]\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Vertical major axis:<\/strong> [latex]\\frac{(x-h)^2}{b^2} + \\frac{(y-k)^2}{a^2} = 1[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">In standard form, the larger denominator tells you the major axis direction. If [latex]a^2[\/latex] is under the [latex]x[\/latex]-term, the major axis is horizontal; if under the [latex]y[\/latex]-term, it&#8217;s vertical.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Converting from general form:<\/strong> Use completing the square on both [latex]x[\/latex] and [latex]y[\/latex] terms separately. Group like terms, factor out coefficients, complete the square for each variable, then divide to get the equation equal to 1.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1167793373866\" data-type=\"problem\">\n<p id=\"fs-id1167793373868\">Put the equation [latex]9{x}^{2}+16{y}^{2}+18x - 64y - 71=0[\/latex] into standard form and graph the resulting ellipse.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558894\">Hint<\/button><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793240741\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167793287354\">Move the constant over and complete the square.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558895\">Show Solution<\/button><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793263801\" data-type=\"solution\">\n<p id=\"fs-id1167793263804\" style=\"text-align: center;\">[latex]\\frac{{\\left(x+1\\right)}^{2}}{16}+\\frac{{\\left(y - 2\\right)}^{2}}{9}=1[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 430px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225348\/CNX_Calc_Figure_11_05_010.jpg\" alt=\"An ellipse is drawn with equation 9x2 + 16y2 + 18x \u2013 64y \u2212 71 = 0. It has center at (\u22121, 2), touches the x-axis at (2, 0) and (\u22124, 0), and touches the y-axis near (0, \u22121) and (0, 5).\" width=\"430\" height=\"347\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 10.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Hyperbolas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A hyperbola is defined by a difference relationship rather than a sum. Every point on a hyperbola maintains a constant difference between its distances to two fixed points called foci. This difference-based definition creates the hyperbola&#8217;s distinctive two-branch shape.<\/p>\n<p class=\"whitespace-normal break-words\">For any point [latex]P[\/latex] on the hyperbola, [latex]|d(P, F_1) - d(P, F_2)| = 2a[\/latex], where [latex]a[\/latex] is the distance from center to vertex. This difference never changes, no matter which point you choose.<\/p>\n<p>Unlike ellipses (which are &#8220;closed&#8221; curves), hyperbolas have two separate branches that extend infinitely, approaching but never touching their asymptotes.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Key relationship:<\/strong> For hyperbolas, [latex]c^2 = a^2 + b^2[\/latex] (note the plus sign, unlike ellipses). This means [latex]c > a[\/latex] always\u2014the foci are farther from the center than the vertices are.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Standard form equations:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Horizontal transverse axis:<\/strong> [latex]\\frac{(x-h)^2}{a^2} - \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Vertical transverse axis:<\/strong> [latex]\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The positive term tells you the transverse axis direction. If [latex]x[\/latex] is positive, the hyperbola opens left-right; if [latex]y[\/latex] is positive, it opens up-down.<\/p>\n<p class=\"whitespace-normal break-words\">Asymptotes are crucial.\u00a0They guide the hyperbola&#8217;s shape. For horizontal hyperbolas: [latex]y = k \\pm \\frac{b}{a}(x-h)[\/latex]. For vertical hyperbolas: [latex]y = k \\pm \\frac{a}{b}(x-h)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Converting from general form:<\/strong> Use completing the square on both variables, but watch the signs carefully. One squared term is positive, one is negative.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1167793979474\" data-type=\"problem\">\n<p id=\"fs-id1167793979476\">Put the equation [latex]4{y}^{2}-9{x}^{2}+16y+18x - 29=0[\/latex] into standard form and graph the resulting hyperbola. What are the equations of the asymptotes?<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558891\">Hint<\/button><\/p>\n<div id=\"q44558891\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794159385\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167794159392\">Move the constant over and complete the square. Check which direction the hyperbola opens.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558892\">Show Solution<\/button><\/p>\n<div id=\"q44558892\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793979522\" data-type=\"solution\">\n<p id=\"fs-id1167793979524\" style=\"text-align: center;\">[latex]\\frac{{\\left(y+2\\right)}^{2}}{9}-\\frac{{\\left(x - 1\\right)}^{2}}{4}=1[\/latex]. This is a vertical hyperbola. Asymptotes [latex]y=-2\\pm \\frac{3}{2}\\left(x - 1\\right)[\/latex]. <span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 454px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225359\/CNX_Calc_Figure_11_05_014.jpg\" alt=\"A hyperbola is drawn with equation 4y2 \u2013 9x2 + 16x + 18y \u2013 29 = 0. It has center at (1, \u22122), and the hyperbolas are open to the top and bottom.\" width=\"454\" height=\"534\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 14.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Eccentricity and Directrix<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Eccentricity provides a single number that tells you exactly which type of conic section you&#8217;re dealing with. It&#8217;s defined as the constant ratio of distance-to-focus over distance-to-directrix for any point on the conic.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The eccentricity formula:<\/strong> [latex]e = \\frac{\\text{distance from point to focus}}{\\text{distance from point to directrix}}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The classification system:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]e = 0[\/latex]: Circle (special case of ellipse)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0 < e < 1[\/latex]: Ellipse<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e = 1[\/latex]: Parabola<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e > 1[\/latex]: Hyperbola<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Computing eccentricity from standard form:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Ellipses and circles:<\/strong> [latex]e = \\frac{c}{a}[\/latex] where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Hyperbolas:<\/strong> [latex]e = \\frac{c}{a}[\/latex] where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Parabolas:<\/strong> [latex]e = 1[\/latex] always<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Directrix locations:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Parabolas:<\/strong> One directrix at distance [latex]p[\/latex] from vertex, opposite the focus<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Ellipses\/Hyperbolas:<\/strong> Two directrices at [latex]x = h \\pm \\frac{a^2}{c}[\/latex] (horizontal) or [latex]y = k \\pm \\frac{a^2}{c}[\/latex] (vertical)<\/li>\n<\/ul>\n<p>Eccentricity measures how &#8220;stretched&#8221; a conic is. A circle ([latex]e = 0[\/latex]) has no stretch. As [latex]e[\/latex] approaches 1, an ellipse becomes more elongated. A parabola ([latex]e = 1[\/latex]) represents the boundary case. Hyperbolas ([latex]e > 1[\/latex]) are &#8220;stretched beyond the breaking point&#8221; into separate branches. Think of eccentricity as measuring how far the conic deviates from being a perfect circle. The closer to 0, the more circular; the farther from 0, the more &#8220;eccentric&#8221; the shape becomes.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1167794138959\" data-type=\"problem\">\n<p id=\"fs-id1167794138961\">Determine the eccentricity of the hyperbola described by the equation<\/p>\n<div id=\"fs-id1167794138965\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{{\\left(y - 3\\right)}^{2}}{49}-\\frac{{\\left(x+2\\right)}^{2}}{25}=1[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558879\">Hint<\/button><\/p>\n<div id=\"q44558879\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794139057\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167794069860\">First find the values of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, then determine <em data-effect=\"italics\">c<\/em> using the equation [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558889\">Show Solution<\/button><\/p>\n<div id=\"q44558889\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794139025\" data-type=\"solution\">\n<p id=\"fs-id1167794139028\" style=\"text-align: center;\">[latex]e=\\frac{c}{a}=\\frac{\\sqrt{74}}{7}\\approx 1.229[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Polar Equations of Conic Sections<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">All conic sections can be written in a single, elegant polar form that directly incorporates their defining geometric properties. This unified approach reveals the deep connections between parabolas, ellipses, and hyperbolas.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The universal polar equation:<\/strong> [latex]r = \\frac{ep}{1 \\pm e\\cos\\theta}[\/latex] or [latex]r = \\frac{ep}{1 \\pm e\\sin\\theta}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">where [latex]e[\/latex] is the eccentricity and [latex]p[\/latex] is the focal parameter (distance from focus to directrix).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Analyzing the equation:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Normalize the denominator:<\/strong> Make the constant term equal to 1 by factoring<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Identify eccentricity:<\/strong> The coefficient of the trig function is [latex]e[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Determine orientation:<\/strong> Cosine means horizontal major axis; sine means vertical major axis<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>What each part tells you:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]e = 0[\/latex]: Circle<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0 < e < 1[\/latex]: Ellipse<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e = 1[\/latex]: Parabola<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e > 1[\/latex]: Hyperbola<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>The focal parameter [latex]p[\/latex]:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Parabolas:<\/strong> [latex]p = 2a[\/latex] (twice the distance from vertex to focus)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Ellipses:<\/strong> [latex]p = \\frac{a(1-e^2)}{e}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Hyperbolas:<\/strong> [latex]p = \\frac{a(e^2-1)}{e}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1167793380150\" data-type=\"problem\">\n<p id=\"fs-id1167793380152\">Identify and create a graph of the conic section described by the equation<\/p>\n<div id=\"fs-id1167793380155\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\frac{4}{1 - 0.8\\sin\\theta }[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558849\">Hint<\/button><\/p>\n<div id=\"q44558849\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167794172253\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1167794172261\">First find the values of <em data-effect=\"italics\">e<\/em> and <em data-effect=\"italics\">p<\/em>, and then create a table of values.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558859\">Show Solution<\/button><\/p>\n<div id=\"q44558859\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1167793380185\" data-type=\"solution\">\n<p id=\"fs-id1167793380187\">Here [latex]e=0.8[\/latex] and [latex]p=5[\/latex]. This conic section is an ellipse.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 424px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/09225411\/CNX_Calc_Figure_11_05_018.jpg\" alt=\"Graph of an ellipse with equation r = 4\/(1 \u2013 0.8 sin\u03b8), center near (0, 11), major axis roughly 22, and minor axis roughly 12.\" width=\"424\" height=\"572\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 18.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":677,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1045"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1045\/revisions"}],"predecessor-version":[{"id":2258,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1045\/revisions\/2258"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/677"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1045\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1045"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1045"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1045"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1045"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}