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<p align="justify"><font face="times new roman,times,serif" size="4"><em>La elipse de ecuación </em></font></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><font face="times new roman,times,serif" size="4"><em><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#elip«/mi»«/math»</span></span></span></span></em></font></span></span></span></span></span></span></p> <p align="justify"><span class="nolink"><span class="nolink"></span></span><font face="times new roman,times,serif" size="4"><em>cuyo gráfico es</em></font></p> <p align="justify"><font face="times new roman,times,serif" size="4"><em>#graf</em></font></p> <p align="justify"><font face="times new roman,times,serif" size="4"><em>tiene como focos y centro a los puntos de coordenadas:</em></font></p>
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<p align="justify"><font face="times new roman,times,serif" size="4"><em>Solución:</em></font></p> <p align="justify"><font face="times new roman,times,serif" size="4"><em>Recordemos que las ecuaciones de las elipses con eje focal horizontal y vertical son, respectivamente:</em></font></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><font face="times new roman,times,serif" size="4"><em><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable columnalign="left" rowspacing="0"»«mtr»«mtd»«mfrac»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»h«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»+«/mo»«mfrac»«msup»«mfenced»«mrow»«mi»y«/mi»«mo»-«/mo»«mi»k«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mn»1«/mn»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»h«/mi»«mi»o«/mi»«mi»r«/mi»«mi»i«/mi»«mi»z«/mi»«mi»o«/mi»«mi»n«/mi»«mi»t«/mi»«mi»a«/mi»«mi»l«/mi»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»h«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»+«/mo»«mfrac»«msup»«mfenced»«mrow»«mi»y«/mi»«mo»-«/mo»«mi»k«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mn»1«/mn»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»v«/mi»«mi»e«/mi»«mi»r«/mi»«mi»t«/mi»«mi»i«/mi»«mi»c«/mi»«mi»a«/mi»«mi»l«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></em></font></span></span></span></p> <p align="justify"><font face="times new roman,times,serif" size="4"><em>donde <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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></span></span></span></span></span> es la coordenada del centro de la elipse, <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»a«/mi»«/math»</span></span></span></span></span></span></span> es la distancia del centro al vértice que pasa por el eje mayor y <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»b«/mi»«/math»</span></span></span></span></span></span></span> es la distancia del centro al vértice que pasa por el eje menor.</em></font></p> <p align="justify"><span class="nolink"><span class="nolink"><font face="times new roman,times,serif" size="4"><em>Las coordenadas del centro de esta elipse son <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»C«/mi»«mo»=«/mo»«mfenced»«mrow»«mi»h«/mi»«mo»,«/mo»«mi»k«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«mi»#h1«/mi»«mo»,«/mo»«mi»#k1«/mi»«/mrow»«/mfenced»«/math»</span></span></span></span></span>.</em></font></span></span></p> <p align="justify"><span class="nolink"></span><font face="times new roman,times,serif"><font size="4"><em>Como <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mi»#a2«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mi»#b2«/mi»«/math»</span></span></span></span></span></span> , entonces <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»a«/mi»«mo»=«/mo»«mi»#a1«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mi»b«/mi»«mo»=«/mo»«mi»#b1«/mi»«/math»</span></span></span></span></span></span>. </span></em></font></font></p> <p align="justify"><span class="nolink"></span><span class="nolink"><font face="times new roman,times,serif" size="4"><em>Las ecuaciones que describen las coordenadas de los focos de la elipse son:</em></font></span></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><font face="times new roman,times,serif" size="4"><em><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable columnalign="left" rowspacing="0"»«mtr»«mtd»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfenced»«mrow»«mi»h«/mi»«mo»-«/mo»«mi»c«/mi»«mo»,«/mo»«mi»k«/mi»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mfenced»«mrow»«mi»h«/mi»«mo»+«/mo»«mi»c«/mi»«mo»,«/mo»«mi»k«/mi»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»e«/mi»«mi»l«/mi»«mi»i«/mi»«mi»p«/mi»«mi»s«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»h«/mi»«mi»o«/mi»«mi»r«/mi»«mi»i«/mi»«mi»z«/mi»«mi»o«/mi»«mi»n«/mi»«mi»t«/mi»«mi»a«/mi»«mi»l«/mi»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfenced»«mrow»«mi»h«/mi»«mo»,«/mo»«mi»k«/mi»«mo»-«/mo»«mi»c«/mi»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mfenced»«mrow»«mi»h«/mi»«mo»,«/mo»«mi»k«/mi»«mo»+«/mo»«mi»c«/mi»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»e«/mi»«mi»l«/mi»«mi»i«/mi»«mi»p«/mi»«mi»s«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»v«/mi»«mi»e«/mi»«mi»r«/mi»«mi»t«/mi»«mi»i«/mi»«mi»c«/mi»«mi»a«/mi»«mi»l«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></em></font></span></span></span></p> <p align="justify"><span class="nolink"><font face="times new roman,times,serif"><font size="4"><em>Para calcular <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»c«/mi»«/math»</span></span></span></span></span> , usamos la siguiente propiedad que se cumple en toda elipse <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»c«/mi»«mn»2«/mn»«/msup»«/math»</span></span></span></span></span>, reemplazamos </span></span></em></font></font></span></p> <p align="justify"><font face="times new roman,times,serif"><font size="4"><em><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#a2«/mi»«mo»=«/mo»«mi»#b2«/mi»«mo»+«/mo»«msup»«mi»c«/mi»«mn»2«/mn»«/msup»«/math»</span></span></span></span></span>, </span></span></span></span></em></font></font></p> <p align="justify"><font face="times new roman,times,serif"><font size="4"><em><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">despejamos y obtenemos que <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»c«/mi»«mo»=«/mo»«mi»#c«/mi»«/math»</span></span></span></span></span>. C</span></span></span></span></span><span class="nolink">omo en este caso la elipse tiene eje focal <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#d3«/mi»«/math»</span></span></span></span></span>, tenemos que <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»#F11«/mi»«mo»=«/mo»«mi»#F1«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F12«/mi»«mo»=«/mo»«mi»#F2«/mi»«/math»</span></span></span></span></span></span></span></em></font></font></p>
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Choice 1
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#sol
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<font size="4"><span style="font-family: times new roman,times,serif; font-style: italic;">¡Muy bien! Sigue así.</span></font>
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#op1
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<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Observa que en el gráfico de la elipse, los focos están sobre una recta <span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#d3«/mi»«/math»</span></span></span>, por lo tanto, los focos se calculan como</font></p> <p style="font-style: italic;"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»#F11«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F12«/mi»«/math»</span></span></span>.</p>
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#op2
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<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Al parecer calculaste mal los valores de las coordenadas del centro; en este caso son: <span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»h«/mi»«mo»=«/mo»«mi»#h1«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»k«/mi»«mo»=«/mo»«mi»#k1«/mi»«/math»</span></span>. Con esos valores reemplazas en las coordenadas del foco.</font></p>
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#op3
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<p style="font-style: italic;"><font size="4"><font size="4"><span style="font-family: times new roman,times,serif;">La elipse tiene como eje focal </span><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#d3«/mi»«/math»</span><span style="font-family: times new roman,times,serif;"> y los focos tienen como coordenada</span></font><span style="font-family: times new roman,times,serif;">s</span> <span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»#F11«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F12«/mi»«/math»</span></span></span></font><span class="nolink"><span class="nolink"><font size="4"><span style="font-family: times new roman,times,serif;">; además, las coordenadas <font size="4">d</font></span></font><font size="4"><span style="font-family: times new roman,times,serif;">el centro son:</span></font> <span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»h«/mi»«mo»=«/mo»«mi»#h1«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»k«/mi»«mo»=«/mo»«mi»#k1«/mi»«/math»</span>.</span></span></span></p>
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Choice 7
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