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<p align="justify"><font face="times new roman,times,serif" size="4"><em>En la siguiente figura se muestra el gráfico de una hipérbola</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"><span class="nolink"><span class="nolink"></span></span></span></span></span></span></em></font></span></span></span></span></span></span></p> <p style="text-align: center;"><span class="nolink"><span class="nolink"></span></span><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>de ecuación:</em></font></p> <p style="text-align: center;"><font face="times new roman,times,serif" 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"><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"><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"><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»#hiper«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></em></font></p> <p align="justify"><em><font face="Times New Roman" size="4">Los focos,</font></em><em><font face="Times New Roman" size="4"> vértices</font></em><em><font face="Times New Roman" size="4"> y ecuaciones de sus asíntotas son: </font></em></p> <p align="justify"><font face="times new roman,times,serif" size="4"><em></em></font></p>
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<p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><em>Solución:</em></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><em>Los siguientes gráficos muestran las hipérbolas con eje focal horizontal y vertical, respectivamente.</em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></span></span></span></span></span></span></span></span></span></span><em> </em></font></p> <p style="text-align: center; font-family: 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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><img title="hipérbola horizontal" style="width: 290px; height: 207px;" alt="hipérbola horizontal" src="http://aula.quinan.cl/file.php/1/hip._horizontal.png" border="0" hspace="0" /></span></span></span></span></span></span></span></span></span></span></span></span></span></span> </em></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><em></em></font></p> <p style="text-align: center; font-family: times new roman,times,serif;"><font size="4"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><em><img title="hipérbola vertical" style="width: 295px; height: 215px;" alt="hipérbola vertical" src="http://aula.quinan.cl/file.php/1/hip._vertical.png" border="0" hspace="0" /></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><em> En este ejercicio, la hipérbola es <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"><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»#tipo«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>. Luego tenemos que </span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable»«mtr»«mtd»«mi»C«/mi»«mi»e«/mi»«mi»n«/mi»«mi»t«/mi»«mi»r«/mi»«mi»o«/mi»«mo»:«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«mi»C«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced»«mrow»«mi»h«/mi»«mo»,«/mo»«mi»k«/mi»«/mrow»«/mfenced»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/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»F«/mi»«mi»o«/mi»«mi mathvariant="normal"»cos«/mi»«mo»:«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#F01«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#F02«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»V«/mi»«mi»é«/mi»«mi»r«/mi»«mi»t«/mi»«mi»i«/mi»«mi»c«/mi»«mi»e«/mi»«mi»s«/mi»«mo»:«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»V«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#V01«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»V«/mi»«mn»2«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#V02«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»A«/mi»«mi»s«/mi»«mi»í«/mi»«mi»n«/mi»«mi»t«/mi»«mi»o«/mi»«mi»t«/mi»«mi»a«/mi»«mi»s«/mi»«mo»:«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»L«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»:«/mo»«/mtd»«mtd»«mi»#as1«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»L«/mi»«mn»2«/mn»«/msub»«/mtd»«mtd»«mo»:«/mo»«/mtd»«mtd»«mi»#as2«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: 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"><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"><span class="nolink">Observemos que de la figura adjunta de 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»#tipo«/mi»«/math»</span></span></span></span></span></span> tenemos lo siguiente:<br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><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"><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"><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"><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"><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»«mo»§nbsp;«/mo»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> distancia del centro a algún vértice que está en el eje focal.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></em></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><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"><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"><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»«mo»:«/mo»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> distancia del centro a algún vértice que está en la recta perpendicular al eje focal.</em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><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"><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"><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»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> distancia del centro a alguno de los focos.</em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><em>La ecuación de la hipérbola de este ejercicio es: </em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p><font size="4" style="font-family: times new roman,times,serif;"><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"><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"><span class="nolink"></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font> <p style="text-align: center; font-family: times new roman,times,serif;"><font size="4"><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"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#hiper«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><em>de la cual podemos encontrar su centro:</em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»c«/mi»«mo»=«/mo»«mi»#center«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><em>luego tenemos: </em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable columnalign="left" rowspacing="0"»«mtr»«mtd»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mi»#a2«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»/«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msqrt»«mo»§nbsp;«/mo»«/msqrt»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mi»#b2«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»/«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msqrt»«mo»§nbsp;«/mo»«/msqrt»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mo»=«/mo»«mi»#a1«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»b«/mi»«mo»=«/mo»«mi»#b1«/mi»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd/»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><em>Para calcular <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"><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»c«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>, utilizamos la relación dada en la hipérbola</em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msup»«mi»c«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><em>reemplazando los valores de <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"><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»a«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> y <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"><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»b«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> calculados, se tiene:</em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable columnalign="left" rowspacing="0"»«mtr»«mtd»«msup»«mi»c«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»#a1«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»#b1«/mi»«mn»2«/mn»«/msup»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»/«/mo»«msqrt»«mo»§nbsp;«/mo»«/msqrt»«/mtd»«/mtr»«mtr»«mtd»«mi»c«/mi»«mo»=«/mo»«msqrt»«mrow»«mi»#a2«/mi»«mo»+«/mo»«mi»#b2«/mi»«/mrow»«/msqrt»«/mtd»«/mtr»«mtr»«mtd»«mi»c«/mi»«mo»=«/mo»«mi»#c1«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><span class="nolink"><em>Como tenemos los valores de <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"><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»b«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>, reemplazamos en las coordenadas de los focos, vértices y en las asíntotas obteniendo:<br /></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><span class="nolink"></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font><font size="4"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable»«mtr»«mtd»«mi»F«/mi»«mi»o«/mi»«mi mathvariant="normal"»cos«/mi»«mo»:«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#F03«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#F04«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»V«/mi»«mi»é«/mi»«mi»r«/mi»«mi»t«/mi»«mi»i«/mi»«mi»c«/mi»«mi»e«/mi»«mi»s«/mi»«mo»:«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«msub»«mi»V«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#V03«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»V«/mi»«mn»2«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#V04«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»A«/mi»«mi»s«/mi»«mi»í«/mi»«mi»n«/mi»«mi»t«/mi»«mi»o«/mi»«mi»t«/mi»«mi»a«/mi»«mi»s«/mi»«mo»:«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»L«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»:«/mo»«/mtd»«mtd»«mi»#d_1«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«mtd»«msub»«mi»L«/mi»«mn»2«/mn»«/msub»«/mtd»«mtd»«mo»:«/mo»«/mtd»«mtd»«mi»#d_2«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p align="justify" style="font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><em>Luego, la solución es:</em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p> <p style="text-align: center; font-family: times new roman,times,serif;"><font size="4"><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"><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"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#sol«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></em></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></p>
One or multiple answers?
Multiple answers allowed
One answer only
Shuffle the choices?
Number the choices?
a., b., c., ...
A., B., C., ...
1., 2., 3., ...
i., ii., iii., ...
I., II., III., ...
No numbering
WIRIS variables
Algorithm
Choice 1
Answer
#sol
Grade
None
100%
90%
83.33333%
80%
75%
70%
66.66667%
60%
50%
40%
33.33333%
30%
25%
20%
16.66667%
14.28571%
12.5%
11.11111%
10%
5%
-5%
-10%
-11.11111%
-12.5%
-14.28571%
-16.66667%
-20%
-25%
-30%
-33.33333%
-40%
-50%
-60%
-66.66667%
-70%
-75%
-80%
-83.33333%
-90%
-100%
Feedback
<font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;">¡Muy bien! Sigue así.</span></font>
Choice 2
Answer
#op1
Grade
None
100%
90%
83.33333%
80%
75%
70%
66.66667%
60%
50%
40%
33.33333%
30%
25%
20%
16.66667%
14.28571%
12.5%
11.11111%
10%
5%
-5%
-10%
-11.11111%
-12.5%
-14.28571%
-16.66667%
-20%
-25%
-30%
-33.33333%
-40%
-50%
-60%
-66.66667%
-70%
-75%
-80%
-83.33333%
-90%
-100%
Feedback
<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Ten en cuenta que esta hipérbola tiene eje focal <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"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#tipo«/mi»«/math»</span></span></span></span></span></span></span></span></span></span> y entonces los focos son:</font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4"><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»«mi»#F13«/mi»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F14«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></font></p> <p></p>
Choice 3
Answer
#op2
Grade
None
100%
90%
83.33333%
80%
75%
70%
66.66667%
60%
50%
40%
33.33333%
30%
25%
20%
16.66667%
14.28571%
12.5%
11.11111%
10%
5%
-5%
-10%
-11.11111%
-12.5%
-14.28571%
-16.66667%
-20%
-25%
-30%
-33.33333%
-40%
-50%
-60%
-66.66667%
-70%
-75%
-80%
-83.33333%
-90%
-100%
Feedback
<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Observa que del gráfico de esta hipérbola se concluye que tiene eje focal <span class="nolink"><span class="nolink"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#tipo«/mi»«/math»</span></span></span>; por lo tanto, los vértices vienen dados por la fórmula:</font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4"><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»V«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»#V13«/mi»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»V«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#V14«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></font></p>
Choice 4
Answer
#op3
Grade
None
100%
90%
83.33333%
80%
75%
70%
66.66667%
60%
50%
40%
33.33333%
30%
25%
20%
16.66667%
14.28571%
12.5%
11.11111%
10%
5%
-5%
-10%
-11.11111%
-12.5%
-14.28571%
-16.66667%
-20%
-25%
-30%
-33.33333%
-40%
-50%
-60%
-66.66667%
-70%
-75%
-80%
-83.33333%
-90%
-100%
Feedback
<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Recuerda que las asíntotas para la hipérbola de eje focal <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#tipo«/mi»«/math»</span> son:</font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mtable columnalign="left" rowspacing="0"»«mtr»«mtd»«mi»#d_11«/mi»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mi»#d_12«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></font></p>
Choice 5
Answer
Grade
None
100%
90%
83.33333%
80%
75%
70%
66.66667%
60%
50%
40%
33.33333%
30%
25%
20%
16.66667%
14.28571%
12.5%
11.11111%
10%
5%
-5%
-10%
-11.11111%
-12.5%
-14.28571%
-16.66667%
-20%
-25%
-30%
-33.33333%
-40%
-50%
-60%
-66.66667%
-70%
-75%
-80%
-83.33333%
-90%
-100%
Feedback
Choice 6
Answer
Grade
None
100%
90%
83.33333%
80%
75%
70%
66.66667%
60%
50%
40%
33.33333%
30%
25%
20%
16.66667%
14.28571%
12.5%
11.11111%
10%
5%
-5%
-10%
-11.11111%
-12.5%
-14.28571%
-16.66667%
-20%
-25%
-30%
-33.33333%
-40%
-50%
-60%
-66.66667%
-70%
-75%
-80%
-83.33333%
-90%
-100%
Feedback
Choice 7
Answer
Grade
None
100%
90%
83.33333%
80%
75%
70%
66.66667%
60%
50%
40%
33.33333%
30%
25%
20%
16.66667%
14.28571%
12.5%
11.11111%
10%
5%
-5%
-10%
-11.11111%
-12.5%
-14.28571%
-16.66667%
-20%
-25%
-30%
-33.33333%
-40%
-50%
-60%
-66.66667%
-70%
-75%
-80%
-83.33333%
-90%
-100%
Feedback
Combined feedback
For any correct response
For any partially correct response
Options
Show the number of correct responses once the question has finished
For any incorrect response
Settings for multiple tries
Penalty for each incorrect try
100%
50%
33.33333%
25%
20%
10%
0%
Hint 1
Hint text
Options
Clear incorrect responses
Show the number of correct responses
Hint 2
Hint text
Options
Clear incorrect responses
Show the number of correct responses
Created / last saved
Created
by
Admin User
on
Tuesday, 17 September 2013, 9:51 AM
Last saved
by
Admin User
on
Tuesday, 17 September 2013, 9:51 AM
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