{"id":1950,"date":"2020-02-02T11:34:51","date_gmt":"2020-02-02T10:34:51","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1950"},"modified":"2024-06-09T16:59:36","modified_gmt":"2024-06-09T14:59:36","slug":"preparer-son-devoir-sur-les-nombres-complexes-partie-geometrie","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/02\/preparer-son-devoir-sur-les-nombres-complexes-partie-geometrie\/","title":{"rendered":"Pr\u00e9parer son devoir sur les nombres complexes, partie G\u00e9om\u00e9trie"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Dans <a rel=\"noreferrer noopener\" aria-label=\"cet article (s\u2019ouvre dans un nouvel onglet)\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/01\/29\/preparer-son-devoir-sur-les-nombres-complexes-partie-algebre\/\" target=\"_blank\">cet article<\/a>, je vous expose des exercices pour vous pr\u00e9parer au devoir sur les nombres complexes, partie Alg\u00e8bre. Dans celui-ci, je vous expose plusieurs exercices tomb\u00e9s au bac S qui vous permettront de vous pr\u00e9parer \u00e0 la seconde partie de ce chapitre : la g\u00e9om\u00e9trie.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/02\/preparer-son-devoir-sur-les-nombres-complexes-partie-geometrie\/#Antilles-Guyanne_septembre_2017_3_points\" >Antilles-Guyanne, septembre 2017 (3 points)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/02\/preparer-son-devoir-sur-les-nombres-complexes-partie-geometrie\/#Pondichery_2017_3_points\" >Pondich\u00e9ry 2017 (3 points)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/02\/preparer-son-devoir-sur-les-nombres-complexes-partie-geometrie\/#Liban_mai_2019\" >Liban, mai 2019<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/02\/preparer-son-devoir-sur-les-nombres-complexes-partie-geometrie\/#Metropole_septembre_2017\" >M\u00e9tropole, septembre 2017<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Antilles-Guyanne_septembre_2017_3_points\"><\/span>Antilles-Guyanne, septembre 2017 (3 points)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Soit la suite de nombres complexes \\(\\left(z_n\\right)\\) d\u00e9finie par:$$\\left\\{\\begin{array}{l c l} z_0&amp; =&amp; 100\\\\ z_{n+1}&amp; =&amp;\\dfrac{\\text{i}}{3}z_n \\text{ pour tout entier naturel }\\:n. \\end{array}\\right.$$Le plan est muni d&rsquo;un rep\u00e8re orthonorm\u00e9 direct \\((O;\\vec{u},\\vec{v})\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour tout entier naturel <em>n<\/em>, on note \\(M_n\\) le point d&rsquo;affixe \\(z_n\\).<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>D\u00e9montrer que, pour tout entier naturel <em>n<\/em>, les points O, \\(M_n\\) et \\(M_{n+2}\\) sont align\u00e9s.<\/li>\n\n\n\n<li>On rappelle qu&rsquo;un disque de centre A et de rayon <em>r<\/em>, o\u00f9 <em>r<\/em> est un nombre r\u00e9el positif, est l&rsquo;ensemble des points <em>M<\/em> du plan tels que \\(AM \\leq r\\). D\u00e9montrer que, \u00e0 partir d&rsquo;un certain rang, tous les points \\(M_n\\) appartiennent au disque de centre O et de rayon 1.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Pondichery_2017_3_points\"><\/span>Pondich\u00e9ry 2017 (3 points)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On munit le plan complexe d&rsquo;un rep\u00e8re orthonorm\u00e9 direct  \\((O;\\vec{u},\\vec{v})\\) .  <\/p>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<ol><li>On consid\u00e8re l&rsquo;\u00e9quation $$(E) :\\quad  z^2 &#8211; 6z + c = 0$$o\u00f9 <em>c<\/em> est un r\u00e9el strictement sup\u00e9rieur \u00e0 9.<\/li><li><ol><li>Justifier que (E) admet deux solutions complexes non r\u00e9elles.<\/li><li>Justifier que les solutions de (E) sont \\(z_{\\text{A}} = 3 + \\text{i}\\sqrt{c &#8211; 9}\\)  et \\(z_{\\text{B}} = 3 &#8211; \\text{i}\\sqrt{c &#8211; 9}\\).<\/li><\/ol><\/li><li> On note A et B les points d&rsquo;affixes respectives \\(z_{\\text{A}}\\) et \\(z_{\\text{B}}\\). Justifier que le triangle OAB est isoc\u00e8le en O.<\/li><li> D\u00e9montrer qu&rsquo;il existe une valeur du r\u00e9el <em>c<\/em> pour laquelle le triangle OAB est rectangle et d\u00e9terminer cette valeur.  <\/li><\/ol>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Liban_mai_2019\"><\/span>Liban, mai 2019<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Le plan complexe est muni d&rsquo;un rep\u00e8re orthonorm\u00e9 direct \\((O;\\vec{u},\\vec{v})\\) d&rsquo;unit\u00e9 2 cm. On appelle <em>f<\/em> la fonction qui, \u00e0 tout point <em>M<\/em>, distinct du point O et d&rsquo;affixe un nombre complexe <em>z<\/em>, associe le point <em>M&rsquo;<\/em>  <br> d&rsquo;affixe <em>z&rsquo;<\/em> tel que:$$z&rsquo; = &#8211; \\frac{1}{z}.$$<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>On consid\u00e8re les points A et B d&rsquo;affixes respectives \\(z_{\\text{A}} = &#8211; 1 + \\text{i}\\) et \\(z_{\\text{B}} = \\dfrac{1}{2} \\text{e}^{\\text{i}\\frac{\\pi}{3}}\\).\n<ol class=\"wp-block-list\">\n<li>D\u00e9terminer la forme alg\u00e9brique de l&rsquo;affixe du point A&rsquo;  image du point A par la fonction <em>f<\/em>. <\/li>\n\n\n\n<li>D\u00e9terminer la forme exponentielle de l&rsquo;affixe du point B&rsquo; image du point B par la fonction <em>f<\/em>.<\/li>\n\n\n\n<li>Sur la copie, placer les points A, B, A&rsquo; et B&rsquo; dans le rep\u00e8re orthonorm\u00e9. Pour les points B et B&rsquo;, on laissera les traits de construction apparents.<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li>Soit <em>r<\/em> un r\u00e9el strictement positif et \\(\\theta\\) un r\u00e9el. On consid\u00e8re le complexe <em>z<\/em> d\u00e9fini par \\(z = r\\text{e}^{\\text{i}\\theta}\\).\n<ol class=\"wp-block-list\">\n<li>Montrer que \\(z&rsquo; = \\frac{1}{r}\\text{e}^{\\text{i}(\\pi &#8211; \\theta)}\\).<\/li>\n\n\n\n<li>Est-il vrai que si un point <em>M<\/em>, distinct de O, appartient au disque de centre O et de rayon 1 sans appartenir au cercle de centre O et de rayon 1, alors son image <em>M\u2019<\/em> par la fonction <em>f<\/em> est \u00e0 l&rsquo;ext\u00e9rieur de ce disque ? Justifier. <\/li>\n<\/ol>\n<\/li>\n\n\n\n<li>Soit le cercle \\(\\Gamma\\) de centre K d&rsquo;affixe \\(z_{\\text{K}} = -\\frac{1}{2}\\)  et de rayon \\(\\frac{1}{2}\\).\n<ol class=\"wp-block-list\">\n<li>Montrer qu&rsquo;une \u00e9quation cart\u00e9sienne du cercle \\(\\Gamma\\) est \\(x^2 + x + y^2 = 0\\).  <\/li>\n\n\n\n<li>Soit \\(z = x + \\text{i}y\\) avec <em>x<\/em> et <em>y<\/em> non tous les deux nuls. D\u00e9terminer la forme alg\u00e9brique de <em>z\u2019<\/em> en fonction de <em>x<\/em> et <em>y<\/em>.<\/li>\n\n\n\n<li>Soit <em>M<\/em> un point, distinct de O, du cercle \\(\\Gamma\\). Montrer que l&rsquo;image <em>M\u2019<\/em> du point <em>M<\/em> par la fonction <em>f<\/em> appartient \u00e0 la droite d&rsquo;\u00e9quation <em>x<\/em> = 1.  <\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Metropole_septembre_2017\"><\/span>M\u00e9tropole, septembre 2017<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Le plan complexe est rapport\u00e9 \u00e0 un rep\u00e8re orthonorm\u00e9  \\((O;\\vec{u},\\vec{v})\\) . \u00c0 tout point <em>M<\/em> d&rsquo;affixe <em>z<\/em>, on associe le point <em>M\u2019<\/em> d&rsquo;affixe:$$z\u2019 = &#8211; z^2 + 2z.$$ Le point <em>M\u2019<\/em> est appel\u00e9 image du point <em>M<\/em>.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>R\u00e9soudre dans l&rsquo;ensemble des nombres complexes l&rsquo;\u00e9quation :$$ z^2 + 2z &#8211; 2 = 0.$$En d\u00e9duire les affixes des points dont l&rsquo;image est le point d&rsquo;affixe 2.<\/li>\n\n\n\n<li>Soit <em>M<\/em> un point d&rsquo;affixe <em>z<\/em> et <em>M\u2019<\/em> son image d&rsquo;affixe <em>z<\/em>&lsquo;. On note <em>N<\/em> le point d&rsquo;affixe \\(z_N = z^2\\).  Montrer que <em>M<\/em> est le milieu du segment [<em>NM\u2019<\/em>]. <\/li>\n\n\n\n<li>Dans cette question, on suppose que le point <em>M<\/em> ayant pour affixe <em>z<\/em>, appartient au cercle \\(\\mathcal{C}\\) de centre O et de rayon 1. On note \\(\\theta\\) un argument de <em>z<\/em>.\n<ol class=\"wp-block-list\">\n<li>D\u00e9terminer le module de chacun des nombres complexes <em>z<\/em> et \\(z_N\\), ainsi qu&rsquo;un argument de \\(z_N\\) en fonction de \\(\\theta\\).<\/li>\n\n\n\n<li>  Sur la figure donn\u00e9e en annexe (voir fin de l&rsquo;exercice), on a repr\u00e9sent\u00e9 un point <em>M<\/em> sur le cercle \\(\\mathcal{C}\\). Construire sur cette figure les points <em>N<\/em> et <em>M&rsquo;<\/em> en utilisant une r\u00e8gle et un compas (on laissera les traits de construction apparents). <\/li>\n\n\n\n<li>Soit A le point d&rsquo;affixe 1. Quelle est la nature du triangle <em>AMM\u2019<\/em> ? <\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"366\" height=\"363\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/annexe-nombres-complexes.png\" alt=\"\" class=\"wp-image-1951\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/annexe-nombres-complexes.png 366w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/annexe-nombres-complexes-300x298.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/annexe-nombres-complexes-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/annexe-nombres-complexes-150x150.png 150w\" sizes=\"auto, (max-width: 366px) 100vw, 366px\" \/><figcaption class=\"wp-element-caption\">Annexe du dernier exercice<\/figcaption><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Vous pourrez trouver les sources \\(\\LaTeX\\) ainsi que le PDF ci-dessous:<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/Nombres-complexes-Partie-2.zip\">T\u00e9l\u00e9charger le fichier ZIP<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/Nombres-complexes-Partie-2.zip\" class=\"wp-block-file__button\" download=\"\">T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Dans cet article, je vous expose des exercices pour vous pr\u00e9parer au devoir sur les nombres complexes, partie Alg\u00e8bre. Dans celui-ci, je vous expose plusieurs exercices tomb\u00e9s au bac S qui vous permettront de vous pr\u00e9parer \u00e0 la seconde partie de ce chapitre : la g\u00e9om\u00e9trie.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[52,163],"class_list":["post-1950","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","tag-geometrie","tag-nombres-complexes"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Pr\u00e9parer son devoir sur les nombres complexes, partie G\u00e9om\u00e9trie - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/02\/preparer-son-devoir-sur-les-nombres-complexes-partie-geometrie\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Pr\u00e9parer son devoir sur les nombres complexes, partie G\u00e9om\u00e9trie - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Dans cet article, je vous expose des exercices pour vous pr\u00e9parer au devoir sur les nombres complexes, partie Alg\u00e8bre. 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