{"id":9048,"date":"2012-05-14T23:10:54","date_gmt":"2012-05-14T22:10:54","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9048"},"modified":"2022-01-14T02:27:42","modified_gmt":"2022-01-14T02:27:42","slug":"quais-sao-os-numeros-complexos-cujos-quadrados-sao-iguais-ao-seu-conjugado","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9048","title":{"rendered":"Quais s\u00e3o os n\u00fameros complexos cujos quadrados s\u00e3o iguais ao seu conjugado?"},"content":{"rendered":"<p><ul id='GTTabs_ul_9048' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9048' class='GTTabs_curr'><a  id=\"9048_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9048' ><a  id=\"9048_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_9048'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Quais s\u00e3o os n\u00fameros complexos cujos quadrados s\u00e3o iguais ao seu conjugado?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9048' onClick='GTTabs_show(1,9048)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9048'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Considerando $z = x + yi$, vem ${z^2} = \\left( {{x^2} &#8211; {y^2}} \\right) + 2xyi$ e $\\overline z\u00a0 = x &#8211; yi$.<\/p>\n<p>Assim, temos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{z^2} = \\overline z }&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{{x^2} &#8211; {y^2} = x} \\\\<br \/>\n{2xy =\u00a0 &#8211; y}<br \/>\n\\end{array}} \\right.} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{{x^2} &#8211; {y^2} = x} \\\\<br \/>\n{y\\left( {2x + 1} \\right) = 0}<br \/>\n\\end{array}} \\right.} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{{x^2} &#8211; {y^2} = x} \\\\<br \/>\n{y = 0 \\vee x =\u00a0 &#8211; \\frac{1}{2}}<br \/>\n\\end{array}} \\right.} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{y = 0} \\\\<br \/>\n{{x^2} = x}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x =\u00a0 &#8211; \\frac{1}{2}} \\\\<br \/>\n{\\frac{1}{4} &#8211; {y^2} =\u00a0 &#8211; \\frac{1}{2}}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{y = 0} \\\\<br \/>\n{x = 0 \\vee x = 1}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x =\u00a0 &#8211; \\frac{1}{2}} \\\\<br \/>\n{y =\u00a0 &#8211; \\frac{{\\sqrt 3 }}{2} \\vee y = \\frac{{\\sqrt 3 }}{2}}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x = 0} \\\\<br \/>\n{y = 0}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x = 1} \\\\<br \/>\n{y = 0}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x =\u00a0 &#8211; \\frac{1}{2}} \\\\<br \/>\n{y =\u00a0 &#8211; \\frac{{\\sqrt 3 }}{2}}<br \/>\n\\end{array}} \\right.}&amp; \\vee &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x =\u00a0 &#8211; \\frac{1}{2}} \\\\<br \/>\n{y = \\frac{{\\sqrt 3 }}{2}}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, s\u00e3o apenas quatro os n\u00fameros complexos cujos quadrados s\u00e3o iguais ao seu conjugado:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z^2} = \\overline z }&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z = 0}&amp; \\vee &amp;{z = 1}&amp; \\vee &amp;{z =\u00a0 &#8211; \\frac{1}{2} &#8211; \\frac{{\\sqrt 3 }}{2}i}&amp; \\vee &amp;{z =\u00a0 &#8211; \\frac{1}{2} + \\frac{{\\sqrt 3 }}{2}i}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":719,\r\n\"height\":445,\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 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