{"id":5505,"date":"2010-11-15T01:35:09","date_gmt":"2010-11-15T01:35:09","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5505"},"modified":"2022-01-12T11:06:45","modified_gmt":"2022-01-12T11:06:45","slug":"uma-circunferencia-e-uma-recta-que-lhe-e-tangente","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5505","title":{"rendered":"Uma circunfer\u00eancia e uma reta que lhe \u00e9 tangente"},"content":{"rendered":"<p><ul id='GTTabs_ul_5505' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5505' class='GTTabs_curr'><a  id=\"5505_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5505' ><a  id=\"5505_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_5505'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Num referencial o. n. $(O,\\vec{i},\\vec{j})$, considere a circunfer\u00eancia de equa\u00e7\u00e3o ${{x}^{2}}+{{y}^{2}}+2x+4y+4=0$.<\/p>\n<ol>\n<li>Determine as coordenadas do centro e o raio da circunfer\u00eancia.<\/li>\n<li>Determine uma equa\u00e7\u00e3o da reta tangente \u00e0 circunfer\u00eancia no ponto $A(0,-2)$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5505' onClick='GTTabs_show(1,5505)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5505'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, \\[\\begin{array}{*{35}{l}}<br \/>\n{{x}^{2}}+{{y}^{2}}+2x+4y+4=0 &amp; \\Leftrightarrow\u00a0 &amp; {{(x+1)}^{2}}-1+{{(y+2)}^{2}}-4+4=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{(x+1)}^{2}}+{{(y+2)}^{2}}=1\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nLogo, a circunfer\u00eancia tem centro $C(-1,-2)$ e raio $r=1$.<br \/>\n\u00ad<\/li>\n<li>Designado por $T(x,y)$ um ponto gen\u00e9rico dessa tangente, tem-se: $\\overrightarrow{AT}.\\overrightarrow{CA}=0$. (Porqu\u00ea?)\n<p>Logo, a reta pretendida pode ser definida por: \\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{AT}.\\overrightarrow{CA}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x,y+2).(1,0)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x=0\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5505' onClick='GTTabs_show(0,5505)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Num referencial o. n. $(O,\\vec{i},\\vec{j})$, considere a circunfer\u00eancia de equa\u00e7\u00e3o ${{x}^{2}}+{{y}^{2}}+2x+4y+4=0$. Determine as coordenadas do centro e o raio da circunfer\u00eancia. Determine uma equa\u00e7\u00e3o da reta tangente \u00e0 circunfer\u00eancia no ponto&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19493,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5505","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-geometria","tag-vectores"],"views":1949,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Tangente_a_uma_circunferencia_2.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5505","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=5505"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5505\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19493"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5505"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5505"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5505"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}