{"id":9373,"date":"2012-06-03T22:20:40","date_gmt":"2012-06-03T21:20:40","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9373"},"modified":"2022-01-14T12:45:32","modified_gmt":"2022-01-14T12:45:32","slug":"identifique-no-conjunto-dos-pontos-do-plano-as-imagens-dos-numeros-complexos-z","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9373","title":{"rendered":"Identifique, no conjunto dos pontos do plano, as imagens dos n\u00fameros complexos $z$"},"content":{"rendered":"<p><ul id='GTTabs_ul_9373' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9373' class='GTTabs_curr'><a  id=\"9373_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9373' ><a  id=\"9373_1\" onMouseOver=\"GTTabsShowLinks('R1'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R1<\/a><\/li>\n<li id='GTTabs_li_2_9373' ><a  id=\"9373_2\" onMouseOver=\"GTTabsShowLinks('R2'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R2<\/a><\/li>\n<li id='GTTabs_li_3_9373' ><a  id=\"9373_3\" onMouseOver=\"GTTabsShowLinks('R3'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R3<\/a><\/li>\n<li id='GTTabs_li_4_9373' ><a  id=\"9373_4\" onMouseOver=\"GTTabsShowLinks('R4'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R4<\/a><\/li>\n<li id='GTTabs_li_5_9373' ><a  id=\"9373_5\" onMouseOver=\"GTTabsShowLinks('R5'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R5<\/a><\/li>\n<li id='GTTabs_li_6_9373' ><a  id=\"9373_6\" onMouseOver=\"GTTabsShowLinks('R6'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R6<\/a><\/li>\n<li id='GTTabs_li_7_9373' ><a  id=\"9373_7\" onMouseOver=\"GTTabsShowLinks('R7'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R7<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_9373'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Identifique, no conjunto dos pontos do plano, as imagens dos n\u00fameros complexos $z$, tais que:<\/p>\n<ol>\n<li>$\\left| {z + 1 + 2i} \\right| = 2$<\/li>\n<li>$\\left| {z &#8211; i + 2} \\right| \\leqslant 3$<\/li>\n<li>$\\left| {z + 2 &#8211; 4i} \\right| = \\left| {2i &#8211; z} \\right|$<\/li>\n<li>$\\left| {\\frac{1}{z}} \\right| &lt; \\frac{1}{4}$<\/li>\n<li>$z.\\overline z\u00a0 = z + \\overline z $<\/li>\n<li>$2\\left| {{\\text{z &#8211; 1}}} \\right| \\leqslant \\left| {{\\text{z + 2}}} \\right|$<\/li>\n<li>$\\operatorname{Im} \\left( {\\frac{1}{{z + 1}}} \\right) \\geqslant 1$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(1,9373)'>R1 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9373'>\n<span class='GTTabs_titles'><b>R1<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$\\left| {z + 1 + 2i} \\right| = 2 \\Leftrightarrow \\left| {z &#8211; \\left( { &#8211; 1 &#8211; 2i} \\right)} \\right| = 2$$<\/p>\n<\/blockquote>\n<p>A condi\u00e7\u00e3o define a circunfer\u00eancia de centro $\\left( { &#8211; 1, &#8211; 2} \\right)$, afixo de ${z_1} =\u00a0 &#8211; 1 &#8211; 2i$, e raio 2 unidades.<\/p>\n<p>Com efeito, considerando $z = x + yi$, vem:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\left| {\\left( {x + yi} \\right) &#8211; \\left( { &#8211; 1 &#8211; 2i} \\right)} \\right| = 2}&amp; \\Leftrightarrow &amp;{\\left| {\\left( {x + 1} \\right) + \\left( {y + 2} \\right)i} \\right| = 2} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\sqrt {{{\\left( {x + 1} \\right)}^2} + {{\\left( {y + 2} \\right)}^2}}\u00a0 = 2} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\left( {x + 1} \\right)}^2} + {{\\left( {y + 2} \\right)}^2} = 4}<br \/>\n\\end{array}$$<\/p>\n<p style=\"text-align: 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i} \\right)} \\right| \\leqslant 3$$<\/p>\n<\/blockquote>\n<p>A condi\u00e7\u00e3o define o c\u00edrculo de centro $\\left( { &#8211; 2,1} \\right)$, afixo de ${z_1} =\u00a0 &#8211; 2 + i$, e raio 3 unidades.<\/p>\n<p>Com efeito, considerando $z = x + yi$, vem:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\left| {\\left( {x + yi} \\right) &#8211; \\left( { &#8211; 2 + i} \\right)} \\right| \\leqslant 3}&amp; \\Leftrightarrow &amp;{\\left| {\\left( {x + 2} \\right) + \\left( {y &#8211; 1} \\right)i} \\right| \\leqslant 3} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\sqrt {{{\\left( {x + 2} \\right)}^2} + {{\\left( {y &#8211; 1} \\right)}^2}}\u00a0 \\leqslant 3} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\left( {x + 2} \\right)}^2} + {{\\left( {y &#8211; 1} \\right)}^2} \\leqslant 9}<br \/>\n\\end{array}$$<\/p>\n<p style=\"text-align: center;\"><div id=\"ggbApplet2\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet2\",\r\n\"width\":654,\r\n\"height\":371,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ \"material_id\":12345,\r\n\"ggbBase64\":\"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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet2 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2')};\r\n<\/script><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(1,9373)'>&lt;&lt; R1<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(3,9373)'>R3 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_3_9373'>\n<span class='GTTabs_titles'><b>R3<\/b><\/span><\/p>\n<blockquote>\n<p>$$\\left| {z + 2 &#8211; 4i} \\right| = \\left| {2i &#8211; z} \\right| \\Leftrightarrow \\left| {z &#8211; \\left( { &#8211; 2 + 4i} \\right)} \\right| = \\left| {z &#8211; 2i} \\right|$$<\/p>\n<\/blockquote>\n<p>Note que $\\left| {2i &#8211; z} \\right| = \\left| {z &#8211; 2i} \\right|$, pois dois n\u00fameros complexos sim\u00e9tricos t\u00eam iguais m\u00f3dulos.<\/p>\n<p>A condi\u00e7\u00e3o define a mediatriz do segmento de reta de extremos $\\left( { &#8211; 2,4} \\right)$ e $\\left( {{\\text{0}}{\\text{,2}}} \\right)$, afixos de ${z_1} =\u00a0 &#8211; 2 + 4i$ e de ${z_2} = 2i$, respetivamente.<\/p>\n<p>Com efeito, considerando $z = x + yi$, vem:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\left| {\\left( {x + yi} \\right) &#8211; \\left( { &#8211; 2 + 4i} \\right)} \\right| = \\left| {\\left( {x + yi} \\right) &#8211; 2i} \\right|}&amp; \\Leftrightarrow &amp;{\\left| {\\left( {x + 2} \\right) + \\left( {y &#8211; 4} \\right)i} \\right| = \\left| {x + \\left( {y &#8211; 2} \\right)i} \\right|} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\sqrt {{{\\left( {x + 2} \\right)}^2} + {{\\left( {y &#8211; 4} \\right)}^2}}\u00a0 = \\sqrt {{x^2} + {{\\left( {y &#8211; 2} \\right)}^2}} } \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{x^2} + 4x + 4 + {y^2} &#8211; 8y + 16 = {x^2} + {y^2} &#8211; 4y + 4} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{4x &#8211; 4y + 16 = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{y = x + 4}<br \/>\n\\end{array}$$<\/p>\n<p style=\"text-align: center;\"><div id=\"ggbApplet3\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet3\",\r\n\"width\":654,\r\n\"height\":371,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet3 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2'); applet3.inject('ggbApplet3')};\r\n<\/script><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(2,9373)'>&lt;&lt; R2<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(4,9373)'>R4 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_4_9373'>\n<span class='GTTabs_titles'><b>R4<\/b><\/span><\/p>\n<blockquote>\n<p>$$\\left| {\\frac{1}{z}} \\right| &lt; \\frac{1}{4} \\Leftrightarrow \\frac{1}{{\\left| z \\right|}} &lt; \\frac{1}{4}$$<\/p>\n<\/blockquote>\n<p>Note que para ${z_2} \\ne 0$, tem-se:<\/p>\n<p>$$\\left| {\\frac{{{z_1}}}{{{z_2}}}} \\right| = \\left| {\\frac{{{\\rho _1}\\operatorname{cis} {\\theta _1}}}{{{\\rho _2}\\operatorname{cis} {\\theta _2}}}} \\right| = \\left| {\\frac{{{\\rho _1}}}{{{\\rho _2}}}\\operatorname{cis} \\left( {{\\theta _1} &#8211; {\\theta _2}} \\right)} \\right| = \\frac{{{\\rho _1}}}{{{\\rho _2}}} = \\frac{{\\left| {{z_1}} \\right|}}{{\\left| {{z_2}} \\right|}}$$<\/p>\n<p>Assim, temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\left| {\\frac{1}{z}} \\right| &lt; \\frac{1}{4}}&amp; \\Leftrightarrow &amp;{\\frac{1}{{\\left| z \\right|}} &lt; \\frac{1}{4}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left| z \\right| &gt; 4}&amp; \\wedge &amp;{z \\ne 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left| z \\right| &gt; 4}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, a condi\u00e7\u00e3o define o exterior da circunfer\u00eancia de centro na origem e raio 4 unidades:<\/p>\n<p>$${x^2} + {y^2} &gt; 16$$<\/p>\n<p style=\"text-align: center;\"><div id=\"ggbApplet4\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet4\",\r\n\"width\":654,\r\n\"height\":371,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet4 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2'); applet3.inject('ggbApplet3'); applet4.inject('ggbApplet4')};\r\n<\/script><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(3,9373)'>&lt;&lt; R3<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(5,9373)'>R5 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_5_9373'>\n<span class='GTTabs_titles'><b>R5<\/b><\/span><\/p>\n<blockquote>\n<p>$$z.\\overline z\u00a0 = z + \\overline z $$<\/p>\n<\/blockquote>\n<p>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{z.\\overline z\u00a0 = z + \\overline z }&amp; \\Leftrightarrow &amp;{{{\\left| {\\text{z}} \\right|}^2} = 2 \\times \\operatorname{Re} \\left( z \\right)}<br \/>\n\\end{array}$$<\/p>\n<p>Assim, considerando $z = x + yi$, temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{x^2} + {y^2} = 2x}&amp; \\Leftrightarrow &amp;{{x^2} &#8211; 2x + {y^2} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\left( {x &#8211; 1} \\right)}^2} &#8211; 1 + {y^2} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\left( {x &#8211; 1} \\right)}^2} + {y^2} = 1}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, a condi\u00e7\u00e3o define a circunfer\u00eancia de centro em $\\left( {1,0} \\right)$ e raio 1 unidade.<\/p>\n<p style=\"text-align: center;\"><div id=\"ggbApplet5\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet5\",\r\n\"width\":654,\r\n\"height\":371,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ \"material_id\":12345,\r\n\"ggbBase64\":\"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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet5 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2'); applet3.inject('ggbApplet3'); applet4.inject('ggbApplet4'); applet5.inject('ggbApplet5')};\r\n<\/script><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(4,9373)'>&lt;&lt; R4<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(6,9373)'>R6 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_6_9373'>\n<span class='GTTabs_titles'><b>R6<\/b><\/span><\/p>\n<blockquote>\n<p>$$2\\left| {{\\text{z &#8211; 1}}} \\right| \\leqslant \\left| {{\\text{z + 2}}} \\right|$$<\/p>\n<p><span style=\"background-color: #ccffff;\">Note que, caso o fator $2$ no 1.\u00ba membro fosse $1$, a condi\u00e7\u00e3o definia um semiplano vertical fechado, determinado pela mediatriz do segmento de reta de extremos $\\left( {1,0} \\right)$ e $\\left( { &#8211; 2,0} \\right)$, afixos de ${z_1} = 1 + 0i$ e de ${z_2} =\u00a0 &#8211; 2 + 0i$, respetivamente.<\/span><\/p>\n<\/blockquote>\n<p>Considerando $z = x + yi$, temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{2\\left| {\\left( {{\\text{x + yi}}} \\right){\\text{ &#8211; 1}}} \\right| \\leqslant \\left| {\\left( {x + yi} \\right){\\text{ + 2}}} \\right|}&amp; \\Leftrightarrow &amp;{2\\left| {\\left( {x &#8211; 1} \\right) + {\\text{y}}i} \\right| \\leqslant \\left| {\\left( {x + 2} \\right) + y{\\text{i}}} \\right|} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{2\\sqrt {{{\\left( {x &#8211; 1} \\right)}^2} + {y^2}}\u00a0 \\leqslant \\sqrt {{{\\left( {x + 2} \\right)}^2} + {y^2}} } \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{4\\left( {{{\\text{x}}^2} &#8211; 2x + 1 + {y^2}} \\right) \\leqslant {x^2} + 4x + 4 + {y^2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{3{x^2} &#8211; 12x + 3{y^2} \\leqslant 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\left( {x &#8211; 2} \\right)}^2} &#8211; 4 + {y^2} \\leqslant 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\left( {x &#8211; 2} \\right)}^2} + {y^2} \\leqslant 4}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, a condi\u00e7\u00e3o define o c\u00edrculo de centro $\\left( {2,0} \\right)$ e raio 2 unidades.<\/p>\n<p style=\"text-align: center;\"><div id=\"ggbApplet6\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet6\",\r\n\"width\":654,\r\n\"height\":371,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ \"material_id\":12345,\r\n\"ggbBase64\":\"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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet6 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2'); applet3.inject('ggbApplet3'); applet4.inject('ggbApplet4'); applet5.inject('ggbApplet5'); applet6.inject('ggbApplet6')};\r\n<\/script><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(5,9373)'>&lt;&lt; R5<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9373' onClick='GTTabs_show(7,9373)'>R7 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_7_9373'>\n<span class='GTTabs_titles'><b>R7<\/b><\/span><\/p>\n<blockquote>\n<p>$$\\operatorname{Im} \\left( {\\frac{1}{{z + 1}}} \\right) \\geqslant 1$$<\/p>\n<\/blockquote>\n<p>Como $$\\begin{array}{*{20}{l}}<br \/>\n{\\operatorname{Im} \\left( {\\frac{1}{{z + 1}}} \\right) \\geqslant 1}&amp; \\Leftrightarrow &amp;{\\operatorname{Im} \\left( {\\frac{{\\overline {z + 1} }}{{{{\\left| {z + 1} \\right|}^2}}}} \\right) \\geqslant 1}<br \/>\n\\end{array}$$ considerando $z = x + yi$, temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\operatorname{Im} \\left( {\\frac{{\\overline {\\left( {x + yi} \\right) + 1} }}{{{{\\left| {\\left( {x + yi} \\right) + 1} \\right|}^2}}}} \\right) \\geqslant 1}&amp; \\Leftrightarrow &amp;{\\operatorname{Im} \\left( {\\frac{{\\left( {x + 1} \\right) &#8211; yi}}{{{{\\left( {x + 1} \\right)}^2} + {y^2}}}} \\right) \\geqslant 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\frac{{ &#8211; y}}{{{{\\left( {x + 1} \\right)}^2} + {y^2}}} \\geqslant 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {x + 1} \\right)}^2} + {y^2} \\leqslant\u00a0 &#8211; y}&amp; \\wedge &amp;{\\left( {x \\ne\u00a0 &#8211; 1 \\wedge y \\ne 0} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {x + 1} \\right)}^2} + {{\\left( {y + \\frac{1}{2}} \\right)}^2} &#8211; \\frac{1}{4} \\leqslant 0}&amp; \\wedge &amp;{\\left( {x \\ne\u00a0 &#8211; 1 \\wedge y \\ne 0} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {x + 1} \\right)}^2} + {{\\left( {y + \\frac{1}{2}} \\right)}^2} \\leqslant \\frac{1}{4}}&amp; \\wedge &amp;{\\left( {x \\ne\u00a0 &#8211; 1 \\wedge y \\ne 0} \\right)}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<p>A condi\u00e7\u00e3o define o circulo de centro $\\left( { &#8211; 1, &#8211; \\frac{1}{2}} \\right)$ e raio ${\\frac{1}{2}}$, com exce\u00e7\u00e3o do ponto $\\left( { 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