{"id":28960,"date":"2021-04-28T13:00:22","date_gmt":"2021-04-28T13:00:22","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28960"},"modified":"2024-09-22T02:00:32","modified_gmt":"2024-09-22T02:00:32","slug":"ecuation-of-ellipses-and-circles","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/ecuation-of-ellipses-and-circles\/","title":{"rendered":"Ecuation of Ellipses and Circles"},"content":{"rendered":"<p><center><\/p>\n<h1>Equation of Ellipses and Circles<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\n   This lesson explains the derivation of the equation of ellipses from their geometric definition, which states that the sum of the distances from any point on the ellipse to two fixed foci is constant. Through a detailed algebraic development, the general equation of ellipses and their canonical form is deduced, as well as the connection between ellipses and circles, showing that a circle is a special case of an ellipse when the semi-axes are equal.<br \/>\n   <\/em><\/p>\n<p>   <strong>Learning Objectives:<\/strong><br \/>\n   At the end of this lesson, the student will be able to:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Deduce<\/strong> the equation of ellipses from their geometric definition.<\/li>\n<li><strong>Recognize<\/strong> both the general form and the canonical form of the ellipse equation.<\/li>\n<\/ol>\n<p>   CONTENT INDEX<br \/>\n   <a href=\"#1\">Geometric Formulation<\/a><br \/>\n   <a href=\"#2\">Derivation of the Ellipse Equation<\/a><br \/>\n   <a href=\"#3\">General Equation of Ellipses<\/a><br \/>\n   <a href=\"#4\">Canonical Equation of Ellipses<\/a><br \/>\n   <a href=\"#5\">Reduction to the Circle Equation<\/a>\n   <\/p>\n<p>   <\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HHiC0bp-Vyc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n   <a name=\"1\"><\/a><\/p>\n<h2>Geometric Formulation<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=133s\" target=\"_blank\" rel=\"noopener\"><strong>To obtain the equation that describes ellipses,<\/strong> <\/a> we must reason, as with <a href=\"https:\/\/toposuranos.com\/la-ecuacion-de-las-parabolas-definiciones-y-propiedades\/\" rel=\"noopener\" target=\"_blank\">parabolas,<\/a> about the geometric meaning of these. An ellipse is the set of all points in the plane such that the sum of the distances between them and two points called foci is always the same.<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-rHroj77w4-o\/YIhoGfTvE_I\/AAAAAAAAFAw\/2Yoa3Q2yrmknQMPObDz8wuyDoOehCug5QCLcBGAsYHQ\/s0\/elipse.PNG\" alt=\"Ellipse\" class=\" aligncenter lazyload\" width=\"338\" height=\"241\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-rHroj77w4-o\/YIhoGfTvE_I\/AAAAAAAAFAw\/2Yoa3Q2yrmknQMPObDz8wuyDoOehCug5QCLcBGAsYHQ\/s0\/elipse.PNG\" alt=\"Ellipse\" class=\" aligncenter lazyload\" width=\"338\" height=\"241\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">That is, it will satisfy:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">d(f_1,p) + d(f_2,p) = constant<\/span>\n<p>   <a name=\"2\"><\/a><\/p>\n<h2>Derivation of the Ellipse Equation<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=311s\" target=\"_blank\" rel=\"noopener\"><strong>From the geometric definition<\/strong><\/a> of ellipses, we can obtain an algebraic expression that describes them. To do this easily, however, we will resort to some simplifications. We will consider, without loss of generality, that the foci are located at <span class=\"katex-eq\" data-katex-display=\"false\">f_1 =(-c,0)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">f_2 =(c,0),<\/span> so that if any point <span class=\"katex-eq\" data-katex-display=\"false\">p=(x,y)<\/span> is part of the ellipse, then it must satisfy that:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x+c)^2 + y^2} + \\sqrt{(x-c)^2 + y^2} = 2a<\/span>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-LtAamnh5D78\/YIiBshjM70I\/AAAAAAAAFA4\/hGiHx6jf_nMOOUHfH-Ywj34TyDJDGEv-wCLcBGAsYHQ\/s0\/ecuacion%2Bde%2Blas%2Belipses.PNG\" alt=\"Equation of Ellipses\" class=\" aligncenter lazyload\" width=\"412\" height=\"333\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-LtAamnh5D78\/YIiBshjM70I\/AAAAAAAAFA4\/hGiHx6jf_nMOOUHfH-Ywj34TyDJDGEv-wCLcBGAsYHQ\/s0\/ecuacion%2Bde%2Blas%2Belipses.PNG\" alt=\"Equation of Ellipses\" class=\" aligncenter lazyload\" width=\"412\" height=\"333\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Where <span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{R}<\/span> is a fixed constant. From this, we can build the following reasoning:<\/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\">\\sqrt{(x+c)^2 + y^2} + \\sqrt{(x-c)^2 + y^2} = 2a<\/span><\/td>\n<td>; Geometric definition of ellipse<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x-c)^2 + y^2} = 2a - \\sqrt{(x+c)^2 + y^2}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(x-c)^2 + \\cancel{y^2} = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + (x+c)^2 + \\cancel{y^2}<\/span><\/td>\n<td>; squaring (1)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(x-c)^2 = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + (x+c)^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\cancel{x^2} -2xc + \\cancel{c^2} = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + \\cancel{x^2} +2xc + \\cancel{c^2} <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">-2xc = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} +2xc <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">4a \\sqrt{(x+c)^2 + y^2} = 4a^2 +4xc = 4(a^2 + xc) <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a \\sqrt{(x+c)^2 + y^2} = a^2 + xc <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a^2 [(x+c)^2 + y^2] = (a^2 + xc)^2 <\/span><\/td>\n<td>; squaring (2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a^2 [x^2 + 2xc + c2 + y^2] = a^4 +2a^2xc + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> a^2 x^2 + \\cancel{2xca^2} + a^2 c2 + a^2 y^2 = a^4 + \\cancel{2a^2xc} + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> a^2 x^2 + a^2 c2 + a^2 y^2 = a^4 + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2 (a^2 - c^2) + a^2 y^2 = a^4 - a^2 c^2 =a^2(a^2-c^2) <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{x^2}{a^2} +\\dfrac{ y^2}{a^2-c^2} = 1 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt a^2 - c^2 =: b^2 <\/span><\/td>\n<td>; The number represented by <span class=\"katex-eq\" data-katex-display=\"false\">b^2<\/span> is positive, as seen from the figure.<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{\\dfrac{x^2}{a^2} +\\dfrac{ y^2}{b^2} = 1}<\/span><\/td>\n<td>; From (3) and (4)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\left(\\dfrac{x}{a}\\right)^2 + \\left(\\dfrac{y}{b}\\right)^2 = 1}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">This is what we call the \u00abellipse equation\u00bb.<\/p>\n<p>   <a name=\"3\"><\/a><\/p>\n<h2>General Equation of Ellipses<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=706s\" target=\"_blank\" rel=\"noopener\"><strong>The equation we just obtained<\/strong><\/a> can be brought to its general form through translation transformations by substituting <span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto (x-h)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto (y-k).<\/span> This leads us to the general form of the ellipse equation:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\left(\\dfrac{x-h}{a}\\right)^2 + \\left(\\dfrac{y-k}{b}\\right)^2 = 1}<\/span>\n<p style=\"text-align: justify;\">This is an ellipse centered at the point <span class=\"katex-eq\" data-katex-display=\"false\">(h,k)<\/span>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-lkxt91FvMTs\/YIiQaL9wpII\/AAAAAAAAFBA\/sUxc6ajd6tcPymC8g4oh3M0l2CTI-xOvgCLcBGAsYHQ\/s0\/elipsegeneral.PNG\" alt=\"general ellipse\" class=\" aligncenter lazyload\" width=\"469\" height=\"373\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-lkxt91FvMTs\/YIiQaL9wpII\/AAAAAAAAFBA\/sUxc6ajd6tcPymC8g4oh3M0l2CTI-xOvgCLcBGAsYHQ\/s0\/elipsegeneral.PNG\" alt=\"general ellipse\" class=\" aligncenter lazyload\" width=\"469\" height=\"373\" \/><\/noscript><br \/>\n   <a name=\"4\"><\/a><\/p>\n<h2>Canonical Equation of Ellipses<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=761s\" target=\"_blank\" rel=\"noopener\"><strong>By doing algebra on this,<\/strong><\/a> we arrive at the canonical equation of ellipses:<\/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\">\\left(\\dfrac{x-h}{a}\\right)^2 + \\left(\\dfrac{y-k}{b}\\right)^2 = 1<\/span><\/td>\n<td>; General ellipse equation<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 (x-h)^2 + a^2(y-k)^2 = a^2 b^2<\/span><\/td>\n<td>; Multiply everything by <span class=\"katex-eq\" data-katex-display=\"false\">a^2b^2<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 [x^2-2xh+h^2] + a^2[y^2-2yk + k^2] = a^2 b^2<\/span><\/td>\n<td>; Expand squares<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2-2hb^2 x + h^2b^2 + a^2 y^2-2ka^2y + k^2a^2 = a^2 b^2<\/span><\/td>\n<td>; Expand parentheses<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2- 2hb^2 x + a^2 y^2-2ka^2y +(h^2b^2 + k^2a^2 - a^2 b^2) = 0 <\/span><\/td>\n<td>; Group constant terms<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">In this last expression, we can make the substitutions <span class=\"katex-eq\" data-katex-display=\"false\">A:=b^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">B:=-2hb^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">C:=a^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">D:=-2ka^2<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">E:=h^2b^2 + k^2a^2 - a^2 b^2.<\/span> Thus, we will see that ellipses will be described by equations 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;\">This is what we call the \u00abCanonical Equation of Ellipses\u00bb.<\/p>\n<p style=\"text-align: justify;\">From these developments, some restrictions on the constants of the canonical equation can be extracted. The most important is that <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> must have the same sign; otherwise, we would no longer be talking about an ellipse but rather a hyperbola. There are more restrictions on the constants of the canonical representation, but discussing them now is not the most efficient; we will go over them in detail when we review the characterization of ellipses and hyperbolas.<\/p>\n<p>   <a name=\"5\"><\/a><\/p>\n<h2>Reduction to the Circle Equation<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=948s\" target=\"_blank\" rel=\"noopener\"><strong>One thing we will review when we talk<\/strong><\/a> about the characterization of ellipses is that the constants <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> in the general equation correspond to the semi-axes of the ellipse. If we take both semi-axes and equalize them, making <span class=\"katex-eq\" data-katex-display=\"false\">a=b=r,<\/span> then the ellipse will become a circle with radius <span class=\"katex-eq\" data-katex-display=\"false\">r.<\/span>\n<h3>General Equation of Circles<\/h3>\n<p style=\"text-align: justify;\">In this way, the general equation of circles is obtained as:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">(x-h)^2 + (y-k)^2 = r^2<\/span>\n<h3>Canonical Equation of Circles<\/h3>\n<p style=\"text-align: justify;\">Similarly, the canonical equation of circles is obtained<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Ax^2 + Bx + Cy^2 + Dy + E = 0<\/span>\n<p style=\"text-align: justify;\">In its canonical form, it coincides with ellipses, as we have seen, since circles, as we have seen, are a particular case of ellipses.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equation of Ellipses and Circles Summary: This lesson explains the derivation of the equation of ellipses from their geometric definition, which states that the sum of the distances from any point on the ellipse to two fixed foci is constant. Through a detailed algebraic development, the general equation of ellipses and their canonical form is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28959,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":36,"footnotes":""},"categories":[583,567],"tags":[],"class_list":["post-28960","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Ecuation of Ellipses and Circles - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn how to deduce the equation of ellipses and their relationship with circles. 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