{"id":29021,"date":"2021-05-05T13:00:46","date_gmt":"2021-05-05T13:00:46","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29021"},"modified":"2024-09-22T04:31:23","modified_gmt":"2024-09-22T04:31:23","slug":"conic-sections-characterization-and-graph-of-parabolas-ellipses-and-hyperbolas","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/conic-sections-characterization-and-graph-of-parabolas-ellipses-and-hyperbolas\/","title":{"rendered":"Conic Sections: Characterization and Graph of Parabolas, Ellipses, and Hyperbolas"},"content":{"rendered":"<p><center><\/p>\n<h1>Conic Sections: Characterization and Graph of Parabolas, Ellipses, and Hyperbolas<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nIn this class, we will review the conic sections (parabolas, ellipses, and hyperbolas), starting with their canonical and general equations. We explain how to identify and characterize each curve, focusing on key elements such as the vertex, focus, and axis of symmetry in parabolas, and the distinction between ellipses and hyperbolas according to the signs of their coefficients.<br \/>\n<\/em><br \/>\n<strong>Learning Objectives:<\/strong><br \/>\nBy the end of this class, the student will be able to:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Recognize<\/strong> the canonical equations of conic sections (parabolas, ellipses, hyperbolas)<\/li>\n<li><strong>Calculate<\/strong> each characteristic of the conic sections: length of semi-axes, focal distance, directrix, etc.<\/li>\n<\/ol>\n<p><strong>CONTENT INDEX<\/strong><br \/>\n<a href=\"#1\">Conic sections<\/a><br \/>\n<a href=\"#2\">Review of Parabolas<\/a><br \/>\n<a href=\"#3\">Review of Ellipses and Hyperbolas<\/a><br \/>\n<a href=\"#4\">Characterization of the Ellipse<\/a><br \/>\n<a href=\"#5\">Characterization of the Hyperbola<\/a><br \/>\n<a href=\"#6\">Solved Exercises<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/d21_9EHUv_M\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Conic Sections<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=126s\" target=\"_blank\" rel=\"noopener\"><strong>Conic sections are<\/strong><\/a> the curves that result from intersecting the surface of a cone with a plane. The family of conic sections includes circles and ellipses, and hyperbolas, all of which we have already studied.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-5eckvpNxzlg\/YJJimcxkMYI\/AAAAAAAAFEA\/dfGTvXblcD4dZXSjpWvonYFN8O0EMNqtwCLcBGAsYHQ\/s0\/curvas-conicas-secciones-cono.png\" alt=\"Conic Sections\" class=\" aligncenter lazyload\" width=\"531\" height=\"272\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-5eckvpNxzlg\/YJJimcxkMYI\/AAAAAAAAFEA\/dfGTvXblcD4dZXSjpWvonYFN8O0EMNqtwCLcBGAsYHQ\/s0\/curvas-conicas-secciones-cono.png\" alt=\"Conic Sections\" class=\" aligncenter lazyload\" width=\"531\" height=\"272\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Now we will review the techniques for recognizing and characterizing each of these curves. We will especially focus on the canonical forms because these are the ones most frequently encountered and reveal the least explicit information. The general equations, on the other hand, reveal almost all the geometric characterization.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Review of Parabolas<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=160s\" target=\"_blank\" rel=\"noopener\"><strong>Every parabola is represented<\/strong><\/a> by an equation of the form<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ax^2 + bx + c,<\/span> with <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>\n<p style=\"text-align: justify;\">In terms of this, we obtained:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Vertex Coordinates: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (x_0, y_0)=\\left( -\\dfrac{b}{2a}, c - \\dfrac{b^2}{4a} \\right)<\/span><\/li>\n<li>Focal Position: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f=\\dfrac{1}{4a}<\/span><\/li>\n<li>Focus Coordinates: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle foco=\\left( -\\dfrac{b}{2a}, c - \\dfrac{b^2}{4a} + f \\right) =\\left( -\\dfrac{b}{2a}, c + \\dfrac{1- b^2}{4a} \\right)<\/span><\/li>\n<li>Equation of the Directrix: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= c - \\dfrac{b^2}{4a} - f = c - \\dfrac{1+b^2}{4a}<\/span><\/li>\n<li>Equation of the Axis of Symmetry: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x= -\\dfrac{b}{2a} <\/span><\/li>\n<li>Intersections with the x-axis (if they exist): <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x_{1,2}= \\dfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} <\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">With this, we have all the necessary information to graph any parabola.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Review of Ellipses and Hyperbolas<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=449s\" target=\"_blank\" rel=\"noopener\"><strong>Ellipses and hyperbolas, as we have seen,<\/strong><\/a> have a canonical expression of the form.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Ax^2 + Bx + Cy^2 + Dy + E = 0<\/span>\n<p style=\"text-align: justify;\">Where <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> are constants different from zero, and from what we have studied, it follows that:<\/p>\n<ul style=\"text-align: justify;\">\n<li>If <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> have the same sign, it is an ellipse.<\/li>\n<li>If <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> have opposite signs, it is a hyperbola.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">To clearly separate both cases, we will write:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x + \\gamma y^2 + \\delta y + \\epsilon = 0<\/span> is an ellipse.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x - \\gamma y^2 + \\delta y + \\epsilon = 0<\/span> is a hyperbola.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Where <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta, \\gamma, \\delta<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> are any real numbers and <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> are always positive. Writing it this way allows us to clearly separate both cases. From this, we can make the following inferences:<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Characterization of the Ellipse<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=552s\" target=\"_blank\" rel=\"noopener\"><strong>Starting from the canonical equation,<\/strong><\/a> we have the following deduction:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td width=\"50\">(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x + \\gamma y^2 + \\delta y + \\epsilon = 0<\/span><\/td>\n<td>; canonical equation of ellipses.<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x^2+ \\dfrac{\\beta}{\\alpha }x\\right) + \\gamma \\left(y^2 + \\dfrac{\\delta}{\\gamma }y\\right) =- \\epsilon<\/span><\/td>\n<td>; factoring and regrouping terms<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2 + \\gamma \\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2 =\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<td>; completing squares and regrouping terms<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} + \\gamma \\dfrac{\\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; dividing everything by <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\dfrac{1}{\\alpha }\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} + \\dfrac{\\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\dfrac{1}{ \\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; rearranging <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{ x - \\left(-\\dfrac{\\beta}{2 \\alpha }\\right)}{\\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 + \\left( \\dfrac{y - \\left(-\\dfrac{\\delta}{2 \\gamma } \\right)}{\\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 = 1<\/span><\/td>\n<td>; restructuring with square roots<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">In the development of this deduction, step (3) is particularly delicate, because if the coefficient <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span> is negative, then the ellipse cannot exist.<\/p>\n<p style=\"text-align: justify;\">Recall that the general equation for ellipses is of the form<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{x-h}{a} \\right)^2 + \\left(\\dfrac{y-k}{b} \\right)^2 = 1<\/span>\n<p style=\"text-align: justify;\">With this last result, we now have a direct relationship between the parameters of the general formula that allows us to reveal all the information hidden in the canonical expression:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Center coordinates: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (h,k) = \\left( -\\dfrac{\\beta}{2\\alpha}, -\\dfrac{\\delta}{2\\gamma}\\right)<\/span><\/li>\n<li>Length of the horizontal semi-axis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle a = \\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<li>Length of the vertical semi-axis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b = \\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">With this, it is now possible to recognize and graph an ellipse directly from its canonical form. Its graph will look like this:<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-aVD7YQ7DfL0\/YJKBv9QXeTI\/AAAAAAAAFEQ\/urCuFtrn-YYBQ_fVSGXsmhMqExFumag-ACLcBGAsYHQ\/s0\/elipse.PNG\" class=\" aligncenter lazyload\" width=\"414\" height=\"291\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-aVD7YQ7DfL0\/YJKBv9QXeTI\/AAAAAAAAFEQ\/urCuFtrn-YYBQ_fVSGXsmhMqExFumag-ACLcBGAsYHQ\/s0\/elipse.PNG\" class=\" aligncenter lazyload\" width=\"414\" height=\"291\" \/><\/noscript><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Characterization of the Hyperbola<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=911s\" target=\"_blank\" rel=\"noopener\"><strong>Reasoning in a completely analogous way,<\/strong><\/a> you can make the complete characterization of hyperbolas starting from the canonical equation. In fact, the analysis is so analogous that I will copy and paste the ellipse analysis and only modify a few parts.<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td width=\"50\">(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x - \\gamma y^2 + \\delta y + \\epsilon = 0<\/span><\/td>\n<td>; canonical equation of hyperbolas.<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x^2+ \\dfrac{\\beta}{\\alpha }x\\right) - \\gamma \\left(y^2 - \\dfrac{\\delta}{\\gamma }y\\right) =- \\epsilon<\/span><\/td>\n<td>; factoring and regrouping terms<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2 - \\gamma \\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2 =\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<td>; completing squares and regrouping terms<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} - \\gamma \\dfrac{\\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; dividing everything by <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} - \\dfrac{\\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; rearranging the terms <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{ x - \\left(-\\dfrac{\\beta}{2 \\alpha }\\right)}{\\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 - \\left( \\dfrac{y - \\left(\\dfrac{\\delta}{2 \\gamma } \\right)}{\\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 = 1<\/span><\/td>\n<td>; restructuring with square roots<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">From this, we now have a direct relationship between the canonical equation and the equation of hyperbolas that will allow us to quickly craft their graph.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\dfrac{x-h}{a} \\right)^2 - \\left(\\dfrac{y-k}{b} \\right)^2 =1 <\/span>\n<p style=\"text-align: justify;\">Now, unlike what is done with ellipses, it is more appropriate here to talk about a \u00abgenerating box\u00bb as we will see in the following figure shown later:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Center coordinates: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (h,k) = \\left( -\\dfrac{\\beta}{2\\alpha}, \\dfrac{\\delta}{2\\gamma}\\right)<\/span><\/li>\n<li>Length of the horizontal semi-axis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle a = \\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<li>Length of the vertical semi-axis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b = \\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-bd0n_BuEFiE\/YJKJ1fPDhMI\/AAAAAAAAFEY\/-QjR2QbycSkKJihjHnwmdIDESYgNDyuBgCLcBGAsYHQ\/s0\/hiperbola.PNG\" alt=\"Hyperbola\" class=\" aligncenter lazyload\" width=\"428\" height=\"305\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-bd0n_BuEFiE\/YJKJ1fPDhMI\/AAAAAAAAFEY\/-QjR2QbycSkKJihjHnwmdIDESYgNDyuBgCLcBGAsYHQ\/s0\/hiperbola.PNG\" alt=\"Hyperbola\" class=\" aligncenter lazyload\" width=\"428\" height=\"305\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">With the results of these analyses, we can now graph any member of the conic sections family without any special difficulty.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Solved Exercises<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/PKjQrcC0HG4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Conic Sections: Characterization and Graph of Parabolas, Ellipses, and Hyperbolas Summary: In this class, we will review the conic sections (parabolas, ellipses, and hyperbolas), starting with their canonical and general equations. We explain how to identify and characterize each curve, focusing on key elements such as the vertex, focus, and axis of symmetry in parabolas, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29020,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[583,567],"tags":[],"class_list":["post-29021","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-and-geometry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Conic Sections: Characterization and Graph of Parabolas, Ellipses, and Hyperbolas - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Conic sections are the curves resulting from the intersection of a cone with a plane, including circles, ellipses, parabolas, and hyperbolas, 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