{"id":28973,"date":"2021-04-28T13:00:20","date_gmt":"2021-04-28T13:00:20","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28973"},"modified":"2024-09-22T02:02:09","modified_gmt":"2024-09-22T02:02:09","slug":"equation-des-ellipses-et-des-cercles","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/equation-des-ellipses-et-des-cercles\/","title":{"rendered":"\u00c9quation des ellipses et des cercles"},"content":{"rendered":"<p><center><\/p>\n<h1>\u00c9quation des ellipses et des cercles<\/h1>\n<p><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\n   Ce cours explique comment obtenir l&#8217;\u00e9quation des ellipses \u00e0 partir de leur d\u00e9finition g\u00e9om\u00e9trique, qui stipule que la somme des distances d&#8217;un point quelconque de l&#8217;ellipse \u00e0 deux foyers fixes est constante. \u00c0 travers un d\u00e9veloppement alg\u00e9brique d\u00e9taill\u00e9, on d\u00e9duit l&#8217;\u00e9quation g\u00e9n\u00e9rale des ellipses et leur forme canonique, ainsi que la relation entre les ellipses et les cercles, en montrant qu&#8217;un cercle est un cas particulier d&#8217;ellipse lorsque les demi-axes sont \u00e9gaux.<br \/>\n   <\/em><\/p>\n<p>   <strong>Objectifs d&#8217;apprentissage :<\/strong><br \/>\n   \u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>D\u00e9river<\/strong> l&#8217;\u00e9quation des ellipses \u00e0 partir de leur d\u00e9finition g\u00e9om\u00e9trique.<\/li>\n<li><strong>Reconna\u00eetre<\/strong> la forme g\u00e9n\u00e9rale et la forme canonique de l&#8217;\u00e9quation des ellipses.<\/li>\n<\/ol>\n<p>   SOMMAIRE<br \/>\n   <a href=\"#1\">Formulation g\u00e9om\u00e9trique<\/a><br \/>\n   <a href=\"#2\">Obtention de l&#8217;\u00e9quation des ellipses<\/a><br \/>\n   <a href=\"#3\">\u00c9quation g\u00e9n\u00e9rale des ellipses<\/a><br \/>\n   <a href=\"#4\">\u00c9quation canonique des ellipses<\/a><br \/>\n   <a href=\"#5\">R\u00e9duction \u00e0 l&#8217;\u00e9quation des cercles<\/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>Formulation g\u00e9om\u00e9trique<\/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>Pour obtenir l&#8217;\u00e9quation d\u00e9crivant les ellipses,<\/strong> <\/a> il faut raisonner comme avec les <a href=\"https:\/\/toposuranos.com\/la-ecuacion-de-las-parabolas-definiciones-y-propiedades\/\" rel=\"noopener\" target=\"_blank\">paraboles,<\/a> sur la signification g\u00e9om\u00e9trique de ces derni\u00e8res. Une ellipse est l&#8217;ensemble de tous les points du plan tels que la somme des distances entre ces points et deux points appel\u00e9s foyers est toujours la m\u00eame.<\/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;\">C&#8217;est-\u00e0-dire, on aura :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">d(f_1,p) + d(f_2,p) = constante<\/span>\n<p>   <a name=\"2\"><\/a><\/p>\n<h2>Obtention de l&#8217;\u00e9quation des ellipses<\/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>\u00c0 partir de la d\u00e9finition g\u00e9om\u00e9trique<\/strong><\/a> des ellipses, on peut obtenir une expression alg\u00e9brique qui les d\u00e9crit. Pour simplifier cela, nous consid\u00e9rerons, sans perte de g\u00e9n\u00e9ralit\u00e9, que les foyers sont situ\u00e9s aux points <span class=\"katex-eq\" data-katex-display=\"false\">f_1 =(-c,0)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">f_2 =(c,0),<\/span> de sorte que si un point quelconque <span class=\"katex-eq\" data-katex-display=\"false\">p=(x,y)<\/span> fait partie de l&#8217;ellipse, on aura :<\/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=\"\u00c9quation des 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=\"\u00c9quation des ellipses\" class=\" aligncenter lazyload\" width=\"412\" height=\"333\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">O\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{R}<\/span> est une constante fixe. \u00c0 partir de cela, nous pouvons construire le raisonnement suivant :<\/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>; D\u00e9finition g\u00e9om\u00e9trique de l&#8217;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>; en \u00e9levant au carr\u00e9 (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>; en \u00e9levant au carr\u00e9 (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>; Le nombre repr\u00e9sent\u00e9 par <span class=\"katex-eq\" data-katex-display=\"false\">b^2<\/span> est positif, comme on le voit sur le sch\u00e9ma.<\/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>; D&#8217;apr\u00e8s (3) et (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;\">Ceci est ce que nous appelons \u00abl&#8217;\u00e9quation des ellipses\u00bb.<\/p>\n<p>   <a name=\"3\"><\/a><\/p>\n<h2>\u00c9quation g\u00e9n\u00e9rale des 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>L&#8217;\u00e9quation que nous venons d&#8217;obtenir<\/strong><\/a> peut \u00eatre amen\u00e9e \u00e0 sa forme g\u00e9n\u00e9rale \u00e0 l&#8217;aide de transformations de translation en substituant <span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto (x-h)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto (y-k).<\/span> Cela nous am\u00e8ne \u00e0 la forme g\u00e9n\u00e9rale de l&#8217;\u00e9quation des ellipses :<\/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;\">C&#8217;est une ellipse centr\u00e9e au 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=\"ellipse g\u00e9n\u00e9rale\" 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=\"ellipse g\u00e9n\u00e9rale\" class=\" aligncenter lazyload\" width=\"469\" height=\"373\" \/><\/noscript><br \/>\n   <a name=\"4\"><\/a><\/p>\n<h2>\u00c9quation canonique des 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>En r\u00e9alisant des op\u00e9rations alg\u00e9briques,<\/strong><\/a> on obtient l&#8217;\u00e9quation canonique des 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>; \u00e9quation g\u00e9n\u00e9rale des ellipses<\/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>; Multiplier tout par <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>; D\u00e9velopper les carr\u00e9s<\/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>; D\u00e9velopper les parenth\u00e8ses<\/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>; Regrouper les termes constants<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Dans cette derni\u00e8re expression, on peut faire les 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> et <span class=\"katex-eq\" data-katex-display=\"false\">E:=h^2b^2 + k^2a^2 - a^2 b^2.<\/span> Ainsi, nous verrons que les ellipses seront d\u00e9crites par des \u00e9quations de la forme :<\/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;\">C&#8217;est ce que nous appelons \u00abl&#8217;\u00e9quation canonique des ellipses\u00bb.<\/p>\n<p style=\"text-align: justify;\">\u00c0 partir de ces d\u00e9veloppements, on peut tirer certaines restrictions sur les constantes de l&#8217;\u00e9quation canonique. La plus importante est que <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> doivent avoir le m\u00eame signe ; sinon, nous parlerons d&#8217;une hyperbole au lieu d&#8217;une ellipse. Il existe d&#8217;autres restrictions sur les constantes de la repr\u00e9sentation canonique, mais nous les aborderons en d\u00e9tail lorsque nous examinerons la caract\u00e9risation des ellipses et des hyperboles.<\/p>\n<p>   <a name=\"5\"><\/a><\/p>\n<h2>R\u00e9duction \u00e0 l&#8217;\u00e9quation des cercles<\/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>Lorsque nous parlerons<\/strong><\/a> de la caract\u00e9risation des ellipses, nous verrons que les constantes <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> de l&#8217;\u00e9quation g\u00e9n\u00e9rale correspondent aux demi-axes de l&#8217;ellipse. Si nous rendons les deux demi-axes \u00e9gaux, en fixant <span class=\"katex-eq\" data-katex-display=\"false\">a=b=r,<\/span> alors l&#8217;ellipse se transformera en un cercle de rayon <span class=\"katex-eq\" data-katex-display=\"false\">r.<\/span>\n<h3>\u00c9quation g\u00e9n\u00e9rale des cercles<\/h3>\n<p style=\"text-align: justify;\">De cette fa\u00e7on, on obtient l&#8217;\u00e9quation g\u00e9n\u00e9rale des cercles :<\/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>\u00c9quation canonique des cercles<\/h3>\n<p style=\"text-align: justify;\">De mani\u00e8re similaire, on obtient l&#8217;\u00e9quation canonique des cercles :<\/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;\">Sous sa forme canonique, elle co\u00efncide avec l&#8217;\u00e9quation des ellipses, car les cercles, comme nous l&#8217;avons vu, sont un cas particulier d&#8217;ellipse.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00c9quation des ellipses et des cercles R\u00e9sum\u00e9 : Ce cours explique comment obtenir l&#8217;\u00e9quation des ellipses \u00e0 partir de leur d\u00e9finition g\u00e9om\u00e9trique, qui stipule que la somme des distances d&#8217;un point quelconque de l&#8217;ellipse \u00e0 deux foyers fixes est constante. \u00c0 travers un d\u00e9veloppement alg\u00e9brique d\u00e9taill\u00e9, on d\u00e9duit l&#8217;\u00e9quation g\u00e9n\u00e9rale des ellipses et leur forme [&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":70,"footnotes":""},"categories":[585,569],"tags":[],"class_list":["post-28973","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebre-et-geometrie","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u00c9quation des ellipses et des cercles - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez \u00e0 d\u00e9duire l&#039;\u00e9quation des ellipses et leur relation avec les cercles. 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