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<font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;">Determina la ecuación de la circunferencia que muestra la siguiente figura:<br /><br /></span></font> <div style="text-align: center;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;">#c1</span></font><br /><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"></span></font></div>
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<font size="4" style="font-family: times new roman,times,serif; font-style: italic;">Solución:<br /><br />La ecuación general de la circunferencia es de la forma:<br /><br /></font> <div style="text-align: center;"><font size="4" style="font-family: times new roman,times,serif; font-style: italic;"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»h«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mrow»«mi»y«/mi»«mo»-«/mo»«mi»k«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/math»</span></font><font size="4" style="font-style: italic;"><br style="font-family: times new roman,times,serif;" /></font> <div style="text-align: left;"><font size="4" style="font-family: times new roman,times,serif; font-style: italic;">Donde <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfenced»«mrow»«mi»h«/mi»«mo»,«/mo»«mi»k«/mi»«/mrow»«/mfenced»«/math»</span> es el centro de la circunferencia y <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»§quot;«/mo»«mi»r«/mi»«mo»§quot;«/mo»«/math»</span> es el radio.</font><font size="4" style="font-style: italic;"><br style="font-family: times new roman,times,serif;" /><span style="font-family: times new roman,times,serif;">Podemos encontrar la ecuación de la circunferencia conociendo un punto cualquiera sobre la circunferencia y el punto que es el centro de esta.</span><br style="font-family: times new roman,times,serif;" /><br style="font-family: times new roman,times,serif;" /><span style="font-family: times new roman,times,serif;">Así, según nuestro gráfico, podemos observar que el centro es </span><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfenced»«mrow»«mi»#h«/mi»«mo»,«/mo»«mi»#k«/mi»«/mrow»«/mfenced»«/math»</span><span style="font-family: times new roman,times,serif;"> y un punto que pasa sobre la circunferencia es </span><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfenced»«mrow»«mi»#h1«/mi»«mo»,«/mo»«mi»#k1«/mi»«/mrow»«/mfenced»«/math»</span><font size="4"><span style="font-family: times new roman,times,serif;">.</span></font></font><font size="4"><font size="4"><span style="font-family: times new roman,times,serif;">(</span></font></font><font size="4"><span style="font-family: times new roman,times,serif;"><span style="font-style: italic;">Se puede escoger cualquier punto que esté sobre la circunferencia</span>)</span><br style="font-family: times new roman,times,serif;" /></font><br /> <div style="text-align: justify;"><font size="4" style="font-family: times new roman,times,serif; font-style: italic;"><font>Entonces, lo que buscamos es el radio <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»§quot;«/mo»«mi»r«/mi»«mo»§quot;«/mo»«/math»</span>. Para esto, reemplazamos en la ecuación de la circunferencia el centro y el punto que pasa sobre la circunferencia.<br /><br /><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable columnalign="left" rowspacing="0"»«mtr»«mtd»«mfenced»«mrow»«mi»h«/mi»«mo»,«/mo»«mi»k«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«mi»#h«/mi»«mo»,«/mo»«mi»#k«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«mi»#h1«/mi»«mo»,«/mo»«mi»#k1«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»</span><br />Reemplazando:<br /></font></font><br /></div> <div style="text-align: center;"><font size="4" style="font-family: times new roman,times,serif; font-style: italic;"><font><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable»«mtr»«mtd»«msup»«mfenced»«mrow»«mi»#h1«/mi»«mo»-«/mo»«mi»#h2«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mrow»«mi»#k1«/mi»«mo»-«/mo»«mi»#k2«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«msup»«mfenced»«mrow»«mi»#h1«/mi»«mi»#c2«/mi»«mi»#h3«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mrow»«mi»#k1«/mi»«mi»#c3«/mi»«mi»#k3«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«msup»«mfenced»«mi»#h4«/mi»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mi»#k4«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»#h5«/mi»«mo»+«/mo»«mi»#k5«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»#s6«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»/«/mo»«/mtd»«mtd»«msqrt»«mo»§nbsp;«/mo»«/msqrt»«/mtd»«/mtr»«mtr»«mtd»«msqrt»«mi»#s6«/mi»«/msqrt»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced close="|" open="|"»«mi»r«/mi»«/mfenced»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»</span></font></font><br /></div><font size="4" style="font-family: times new roman,times,serif; font-style: italic;"><font>Como el radio es una distancia, nos interesa sólo el valor positivo de <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mo»§quot;«/mo»«mi»r«/mi»«mo»§quot;«/mo»«/math»</span>, así:<br /><br /></font></font> <div style="text-align: center;"><font size="4" style="font-family: times new roman,times,serif; font-style: italic;"><font><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»r«/mi»«mo»=«/mo»«msqrt»«mi»#s6«/mi»«/msqrt»«/math»</span></font></font><br /> <div style="text-align: left;"><span style="font-style: italic;"><font size="4"><span style="font-family: times new roman,times,serif;">Entonces, <br /></span></font></span> <div style="text-align: center;"><span style="font-style: italic;"></span><span style="font-style: italic;"><font size="4"><span style="font-family: times new roman,times,serif;"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable»«mtr»«mtd»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»#h2«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mrow»«mi»y«/mi»«mo»-«/mo»«mi»#k2«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«msqrt»«mi»#s6«/mi»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«/math»</span></span></font></span><br /><span style="font-style: italic;"><font size="4"><span style="font-family: times new roman,times,serif;"></span></font></span></div></div></div><br /> <font size="4"><span style="font-family: times new roman,times,serif; font-style: italic;">Por lo tanto, la ecuación de la circunferencia de centro <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#P1«/mi»«/math»</span></span></font> <font size="4" style="font-family: times new roman,times,serif; font-style: italic;">y radio <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msqrt»«mi»#s6«/mi»«/msqrt»«/math»</span></font><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"> es:</span></font><br /><br /> <div style="text-align: center;"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable»«mtr»«mtd»«msup»«mfenced»«mrow»«mi»x«/mi»«mi»#c2«/mi»«mi»#h3«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mrow»«mi»y«/mi»«mi»#c3«/mi»«mi»#k3«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mi»#s7«/mi»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«/math»</span><br /></div> </div></div>
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Choice 1
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#sol
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<font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;">¡Muy bien! Sigue así.</span></font>
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#n1
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<font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;">Recuerda que en la fórmula de la circunferencia el radio es al cuadrado.</span></font>
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#n2
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<font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;">Recuerda que el centro de la circunferencia es </span><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mfenced»«mrow»«mi»#h«/mi»«mo»,«/mo»«mi»#k«/mi»«/mrow»«/mfenced»«/math»</span><span style="font-family: times new roman,times,serif;">.</span></font>
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#n3
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<font size="4"><span style="font-family: times new roman,times,serif; font-style: italic;">Al parecer estás calculando mal el radio de la circunferencia.</span></font>
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