{"id":5861,"date":"2010-11-20T23:40:46","date_gmt":"2010-11-20T23:40:46","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5861"},"modified":"2022-01-21T18:32:07","modified_gmt":"2022-01-21T18:32:07","slug":"um-dominio-plano","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5861","title":{"rendered":"Um dom\u00ednio plano"},"content":{"rendered":"<p><ul id='GTTabs_ul_5861' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5861' class='GTTabs_curr'><a  id=\"5861_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5861' ><a  id=\"5861_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_5861'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5864\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5864\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg\" data-orig-size=\"289,371\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"pag182-33\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg\" class=\"alignright wp-image-5864\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33-233x300.jpg\" alt=\"\" width=\"240\" height=\"308\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33-233x300.jpg 233w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33-116x150.jpg 116w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg 289w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Na figura est\u00e1 representado um referencial o. m. Oxy.<\/p>\n<ul>\n<li>A circunfer\u00eancia de centro C \u00e9 tangente ao eixo das ordenadas e \u00e0 reta t, em T.<\/li>\n<li>O ponto C tem coordenadas (-5,2).<\/li>\n<li>A abcissa de T \u00e9 -9.<\/li>\n<\/ul>\n<ol>\n<li>Prove que a ordenada de T \u00e9 5.<\/li>\n<li>Prove que a equa\u00e7\u00e3o reduzida da reta t \u00e9 $y=\\frac{4}{3}x+17$.<\/li>\n<li>Determine a amplitude dos \u00e2ngulos agudos do tri\u00e2ngulo [ABO] e apresente o resultado aproximado \u00e0s cent\u00e9simas.<\/li>\n<li>Escreva uma condi\u00e7\u00e3o que defina a regi\u00e3o colorida da figura (contorno inclu\u00eddo).<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5861' onClick='GTTabs_show(1,5861)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5861'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5864\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5864\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg\" data-orig-size=\"289,371\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"pag182-33\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg\" class=\"alignright wp-image-5864\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33-233x300.jpg\" alt=\"\" width=\"240\" height=\"308\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33-233x300.jpg 233w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33-116x150.jpg 116w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag182-33.jpg 289w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Como a circunfer\u00eancia \u00e9 tangente ao eixo das ordenadas, esse ponto de tang\u00eancia \u00e9 $R(0,2)$.\n<p>Assim, o raio da circunfer\u00eancia \u00e9 $r=\\overline{CR}=5$.<\/p>\n<p>Como T \u00e9 outro ponto da circunfer\u00eancia, ent\u00e3o:<\/p>\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n\\overline{TC}=5 &amp; \\Leftrightarrow\u00a0 &amp; \\sqrt{{{(-5+9)}^{2}}+{{(2-y)}^{2}}}=5\\wedge y&gt;0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{(y-2)}^{2}}+16=25\\wedge y&gt;0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{(y-2)}^{2}}=9\\wedge y&gt;0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; (y=5\\vee y=-1)\\wedge y&gt;0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; y=5\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<p>Logo, a ordenada de T \u00e9 5.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Designado por $P(x,y)$ um ponto gen\u00e9rico da reta t, tem-se $\\overrightarrow{TP}.\\overrightarrow{TC}=0$, pois uma reta tangente a uma circunfer\u00eancia \u00e9 perpendicular ao raio nesse ponto de tang\u00eancia.\n<p>Assim, vem: \\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{TP}.\\overrightarrow{TC}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x+9,y-5).(-5+9;2-5)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 4x+36-3y+15=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; y=\\frac{4}{3}x+17\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Logo, a equa\u00e7\u00e3o reduzida da reta t \u00e9 $y=\\frac{4}{3}x+17$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A amplitude do \u00e2ngulo ABO \u00e9 igual \u00e0 inclina\u00e7\u00e3o da reta t.<br \/>\nLogo, \\[O\\hat{A}B=t{{g}^{1}}({{m}_{t}})=t{{g}^{1}}(\\frac{4}{3})\\simeq 53,13{}^\\text{o}\\]<\/p>\n<p>Como os \u00e2ngulos OBA e OAB s\u00e3o complementares, ser\u00e1 $O\\hat{A}B=90{}^\\text{o}-O\\hat{A}B\\simeq 36,87{}^\\text{o}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Uma condi\u00e7\u00e3o que define a regi\u00e3o colorida da figura (contorno inclu\u00eddo) \u00e9: \\[\\begin{matrix}<br \/>\n{{(x+5)}^{2}}+{(y-2)}^{2}\\ge 25 &amp; \\wedge\u00a0 &amp; y\\le \\frac{4}{3}x+17 &amp; \\wedge\u00a0 &amp; x\\le 0 &amp; \\wedge\u00a0 &amp; y\\ge 0\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5861' onClick='GTTabs_show(0,5861)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado um referencial o. m. Oxy. A circunfer\u00eancia de centro C \u00e9 tangente ao eixo das ordenadas e \u00e0 reta t, em T. O ponto C tem coordenadas&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20828,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,116,67],"series":[],"class_list":["post-5861","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-dominio-plano","tag-geometria"],"views":3927,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/11V1Pag182-33_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5861","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=5861"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5861\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20828"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5861"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5861"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5861"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5861"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}