{"id":29033,"date":"2021-05-05T13:00:45","date_gmt":"2021-05-05T13:00:45","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29033"},"modified":"2024-09-22T04:33:07","modified_gmt":"2024-09-22T04:33:07","slug":"sections-coniques-caracterisation-et-graphiques-des-paraboles-ellipses-et-hyperboles","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/sections-coniques-caracterisation-et-graphiques-des-paraboles-ellipses-et-hyperboles\/","title":{"rendered":"Sections Coniques : Caract\u00e9risation et Graphiques des Paraboles, Ellipses et Hyperboles"},"content":{"rendered":"<p><center><\/p>\n<h1>Sections Coniques : Caract\u00e9risation et Graphiques des Paraboles, Ellipses et Hyperboles<\/h1>\n<p><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nDans ce cours, nous allons revoir les sections coniques (paraboles, ellipses et hyperboles), en commen\u00e7ant par leurs \u00e9quations canoniques et g\u00e9n\u00e9rales. Nous expliquerons comment identifier et caract\u00e9riser chaque courbe, en nous concentrant sur des \u00e9l\u00e9ments cl\u00e9s tels que le sommet, le foyer et l&#8217;axe de sym\u00e9trie pour les paraboles, et la distinction entre les ellipses et les hyperboles selon les signes de leurs coefficients.<br \/>\n<\/em><br \/>\n<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>Reconna\u00eetre<\/strong> les \u00e9quations canoniques des sections coniques (paraboles, ellipses, hyperboles)<\/li>\n<li><strong>Calculer<\/strong> chacune des caract\u00e9ristiques des sections coniques : longueur des demi-axes, distance focale, directrice, etc.<\/li>\n<\/ol>\n<p><strong>TABLE DES MATI\u00c8RES<\/strong><br \/>\n<a href=\"#1\">Sections coniques<\/a><br \/>\n<a href=\"#2\">Revue des Paraboles<\/a><br \/>\n<a href=\"#3\">Revue des Ellipses et Hyperboles<\/a><br \/>\n<a href=\"#4\">Caract\u00e9risation de l&#8217;ellipse<\/a><br \/>\n<a href=\"#5\">Caract\u00e9risation de l&#8217;hyperbole<\/a><br \/>\n<a href=\"#6\">Exercices R\u00e9solus<\/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>Sections coniques<\/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>Les sections coniques sont<\/strong><\/a> toutes les courbes r\u00e9sultant de l&#8217;intersection de la surface d&#8217;un c\u00f4ne avec un plan. La famille des sections coniques comprend les cercles et ellipses, ainsi que les hyperboles, que nous avons d\u00e9j\u00e0 \u00e9tudi\u00e9es.<\/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=\"Sections Coniques\" 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=\"Sections Coniques\" class=\" aligncenter lazyload\" width=\"531\" height=\"272\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Nous allons maintenant examiner les techniques permettant de reconna\u00eetre et de caract\u00e9riser chacune de ces courbes. Nous nous concentrerons en particulier sur les formes canoniques, car elles sont les plus fr\u00e9quentes et r\u00e9v\u00e8lent le moins d&#8217;informations de mani\u00e8re explicite. Les \u00e9quations g\u00e9n\u00e9rales, en revanche, d\u00e9voilent presque toutes les caract\u00e9ristiques g\u00e9om\u00e9triques.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Revue des Paraboles<\/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>Toute parabole est repr\u00e9sent\u00e9e<\/strong><\/a> par une \u00e9quation de la forme<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ax^2 + bx + c,<\/span> avec <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>\n<p style=\"text-align: justify;\">D&#8217;apr\u00e8s cela, nous avons obtenu :<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordonn\u00e9es du sommet : <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>Distance focale : <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f=\\dfrac{1}{4a}<\/span><\/li>\n<li>Coordonn\u00e9es du foyer : <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>\u00c9quation de la directrice : <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>\u00c9quation de l&#8217;axe de sym\u00e9trie : <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x= -\\dfrac{b}{2a} <\/span><\/li>\n<li>Points d&#8217;intersection avec l&#8217;axe des x (si existants) : <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;\">Avec ces informations, nous disposons de tout ce qu&#8217;il faut pour tracer n&#8217;importe quelle parabole.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Revue des Ellipses et Hyperboles<\/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>Les ellipses et hyperboles, comme nous l&#8217;avons vu,<\/strong><\/a> ont une expression canonique de la forme suivante.<\/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;\">O\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> sont des constantes non nulles, et d&#8217;apr\u00e8s ce que nous avons \u00e9tudi\u00e9, il en ressort que :<\/p>\n<ul style=\"text-align: justify;\">\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> ont le m\u00eame signe, il s&#8217;agit d&#8217;une ellipse.<\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> ont des signes oppos\u00e9s, il s&#8217;agit d&#8217;une hyperbole.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Pour bien distinguer les deux cas, nous \u00e9crirons :<\/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> est une 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> est une hyperbole.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">O\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta, \\gamma, \\delta<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> sont des nombres r\u00e9els quelconques, et <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> sont toujours positifs. Cette \u00e9criture permet de distinguer clairement les deux cas. \u00c0 partir de l\u00e0, nous pouvons faire les inf\u00e9rences suivantes :<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Caract\u00e9risation de l&#8217;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>En partant de l&#8217;\u00e9quation canonique,<\/strong><\/a> nous obtenons la d\u00e9duction suivante :<\/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>; \u00e9quation canonique des 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>; factorisation et r\u00e9organisation des termes<\/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>; compl\u00e9tion du carr\u00e9 et r\u00e9organisation des termes<\/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>; en divisant tout par <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>; r\u00e9organisation de <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <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>; r\u00e9organisation avec racines carr\u00e9es<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Dans le d\u00e9veloppement de cette d\u00e9duction, l&#8217;\u00e9tape (3) est particuli\u00e8rement d\u00e9licate, car si le coefficient <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span> est n\u00e9gatif, alors l&#8217;ellipse ne peut pas exister.<\/p>\n<p style=\"text-align: justify;\">Rappelons que l&#8217;\u00e9quation g\u00e9n\u00e9rale des ellipses est de la forme suivante :<\/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;\">Avec ce dernier r\u00e9sultat, nous avons maintenant une relation directe entre les param\u00e8tres de la formule g\u00e9n\u00e9rale qui nous permet de r\u00e9v\u00e9ler toutes les informations cach\u00e9es dans l&#8217;expression canonique :<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordonn\u00e9es du centre : <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>Longueur du demi-axe horizontal : <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>Longueur du demi-axe vertical : <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;\">Avec cela, il est maintenant possible de reconna\u00eetre et de tracer une ellipse directement \u00e0 partir de sa forme canonique. Son graphique ressemblera \u00e0 ce qui suit :<\/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>Caract\u00e9risation de l&#8217;hyperbole<\/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>En raisonnant de mani\u00e8re compl\u00e8tement analogue,<\/strong><\/a> vous pouvez, \u00e0 partir de l&#8217;\u00e9quation canonique, faire la caract\u00e9risation compl\u00e8te des hyperboles. En fait, l&#8217;analyse est tellement analogue que je vais copier et coller l&#8217;analyse des ellipses, en ne modifiant que quelques parties.<\/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>; \u00e9quation canonique des hyperboles.<\/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>; factorisation et r\u00e9organisation des termes<\/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>; compl\u00e9tion du carr\u00e9 et r\u00e9organisation des termes<\/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>; en divisant tout par <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>; r\u00e9organisation des termes <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <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>; r\u00e9organisation avec racines carr\u00e9es<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\u00c0 partir de cela, nous avons maintenant une relation directe entre l&#8217;\u00e9quation canonique et l&#8217;\u00e9quation des hyperboles qui nous permettra de tracer rapidement son graphique.<\/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;\">Contrairement aux ellipses, il est plus correct ici de parler de \u00ab bo\u00eete g\u00e9n\u00e9ratrice \u00bb, comme nous le verrons sur la figure suivante :<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordonn\u00e9es du centre : <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>Longueur du demi-axe horizontal : <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>Longueur du demi-axe vertical : <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=\"Hyperbole\" 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=\"Hyperbole\" class=\" aligncenter lazyload\" width=\"428\" height=\"305\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Avec les r\u00e9sultats de ces analyses, nous pouvons d\u00e9sormais tracer sans difficult\u00e9 n&#8217;importe quel membre de la famille des sections coniques.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Exercices R\u00e9solus<\/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>Sections Coniques : Caract\u00e9risation et Graphiques des Paraboles, Ellipses et Hyperboles R\u00e9sum\u00e9 : Dans ce cours, nous allons revoir les sections coniques (paraboles, ellipses et hyperboles), en commen\u00e7ant par leurs \u00e9quations canoniques et g\u00e9n\u00e9rales. Nous expliquerons comment identifier et caract\u00e9riser chaque courbe, en nous concentrant sur des \u00e9l\u00e9ments cl\u00e9s tels que le sommet, le foyer [&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":8,"footnotes":""},"categories":[585,569],"tags":[],"class_list":["post-29033","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Sections Coniques : Caract\u00e9risation et Graphiques des Paraboles, Ellipses et Hyperboles - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Les sections coniques sont des courbes r\u00e9sultant de l&#039;intersection d&#039;un c\u00f4ne avec un plan, comprenant 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