{"id":6539,"date":"2011-02-28T12:26:56","date_gmt":"2011-02-28T12:26:56","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6539"},"modified":"2021-12-26T03:14:04","modified_gmt":"2021-12-26T03:14:04","slug":"uma-parabola-e-uma-hiperbole","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6539","title":{"rendered":"Uma par\u00e1bola e uma hip\u00e9rbole"},"content":{"rendered":"<p><ul id='GTTabs_ul_6539' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6539' class='GTTabs_curr'><a  id=\"6539_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6539' ><a  id=\"6539_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_6539'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere, num referencial o.n. do plano, os pontos: $A(1,0)$, $B(-1,-1)$ e $C(-3,2)$.<\/p>\n<ol>\n<li>Determine os n\u00fameros reais a, b e c de modo que a par\u00e1bola P, de equa\u00e7\u00e3o $y=a{{x}^{2}}+bx+c$, passe pelos pontos A, B e C.<\/li>\n<li>Considere a hip\u00e9rbole H de equa\u00e7\u00e3o $y=\\frac{1}{x}$.\n<p>a) Verifique que H passa por B.<\/p>\n<p>b) Determine os pontos comuns a P e H.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6539' onClick='GTTabs_show(1,6539)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6539'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Se os pontos A, B e C pertencem \u00e0 par\u00e1bola P, ent\u00e3o as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o da par\u00e1bola:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\na+b+c=0\u00a0 \\\\<br \/>\na-b+c=-1\u00a0 \\\\<br \/>\n9a-3b+c=2\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2b=1\u00a0 \\\\<br \/>\na+b+c=0\u00a0 \\\\<br \/>\n9a-3b+c=2\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=\\frac{1}{2}\u00a0 \\\\<br \/>\na+c=-\\frac{1}{2}\u00a0 \\\\<br \/>\n9a+c=\\frac{7}{2}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=\\frac{1}{2}\u00a0 \\\\<br \/>\n8a=4\u00a0 \\\\<br \/>\na+c=-\\frac{1}{2}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=\\frac{1}{2}\u00a0 \\\\<br \/>\nb=\\frac{1}{2}\u00a0 \\\\<br \/>\nc=-1\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<li>a) H passa por B, pois as coordenadas de B verificam a equa\u00e7\u00e3o de H: $-1=\\frac{1}{-1}\\Leftrightarrow -1=-1$.\n<\/p>\n<p>b) \\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\ny=\\frac{1}{2}{{x}^{2}}+\\frac{1}{2}x-1\u00a0 \\\\<br \/>\ny=\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\frac{1}{2}{{x}^{2}}+\\frac{1}{2}x-1=\\frac{1}{x}\u00a0 \\\\<br \/>\ny=\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n{{x}^{2}}+x-2-\\frac{2}{x}=0\u00a0 \\\\<br \/>\ny=\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\frac{{{x}^{3}}+{{x}^{2}}-2x-2}{x}=0\u00a0 \\\\<br \/>\ny=\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\frac{(x+1)({{x}^{2}}-2)}{x}=0\u00a0 \\\\<br \/>\ny=\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\begin{array}{*{35}{l}}<br \/>\nx=-1 &amp; \\vee\u00a0 &amp; x=\\pm \\sqrt{2}\u00a0 \\\\<br \/>\n\\end{array}\u00a0 \\\\<br \/>\ny=\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {} &amp; {} &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Os pontos comuns a P e H t\u00eam de coordenadas $(-1,-1)$, $(-\\sqrt{2},-\\frac{\\sqrt{2}}{2})$ e $(\\sqrt{2},\\frac{\\sqrt{2}}{2})$.<\/p>\n<p><strong>C\u00e1lculos auxiliares<\/strong>:<\/p>\n<p>Tenha em considera\u00e7\u00e3o que H passa por B.<\/p>\n<p>$\\begin{matrix}<br \/>\n{} &amp; 1 &amp; 1 &amp; -2 &amp; -2\u00a0 \\\\<br \/>\n-1 &amp; {} &amp; -1 &amp; 0 &amp; 2\u00a0 \\\\<br \/>\n{} &amp; 1 &amp; 0 &amp; -2 &amp; 0\u00a0 \\\\<br \/>\n\\end{matrix}$<\/p>\n<p>(Regra de Ruffini)<\/p>\n<p>Resolvendo graficamente, temos:<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6541\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6541\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-1.jpg\" data-orig-size=\"198,134\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"G1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-1.jpg\" class=\"alignnone size-full wp-image-6541\" title=\"G1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-1.jpg\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-1.jpg 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-1-150x101.jpg 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6542\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6542\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-2.jpg\" data-orig-size=\"198,134\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"G2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-2.jpg\" class=\"alignnone size-full wp-image-6542\" title=\"G2\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-2.jpg\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-2.jpg 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-2-150x101.jpg 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-3.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6543\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6543\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-3.jpg\" data-orig-size=\"198,134\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"G3\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-3.jpg\" class=\"alignnone size-full wp-image-6543\" title=\"G3\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-3.jpg\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-3.jpg 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/11-pag187-20-3-150x101.jpg 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a><\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6539' onClick='GTTabs_show(0,6539)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere, num referencial o.n. do plano, os pontos: $A(1,0)$, $B(-1,-1)$ e $C(-3,2)$. Determine os n\u00fameros reais a, b e c de modo que a par\u00e1bola P, de equa\u00e7\u00e3o $y=a{{x}^{2}}+bx+c$, passe pelos&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19411,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,125],"tags":[131],"series":[],"class_list":["post-6539","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-racionais","tag-funcoes-racionais-2"],"views":2775,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/02\/Conicas.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6539","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6539"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6539\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19411"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6539"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6539"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6539"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6539"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}