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<p align="justify"><em><font face="times new roman,times,serif" size="4">Dado el gráfico de la siguiente elipse</font></em></p> <p align="justify"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif" size="4">#graf</font></em></span></span></p> <p align="justify"><em><font face="times new roman,times,serif" size="4">cuyos vértices son los puntos <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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»#V1«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»#V2«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»#V3«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mi»#V4«/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>, entonces, tiene como centro y focos a los puntos:</font></em></p>
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<p align="justify"><em><font face="times new roman,times,serif" size="4">Solución:</font></em></p> <p align="justify"><em><font face="Times New Roman" size="4">Observemos los gráficos de las elipses de eje focal horizontal y vertical:</font></em></p> <p align="justify"><em><font face="Times New Roman" size="4"><img width="560" height="378" title="elipse horizontal" style="width: 560px; height: 378px;" alt="elipse horizontal" src="http://aula.quinan.cl/file.php/1/elip_hor.PNG" border="0" hspace="0" /> <img width="502" height="449" title="Elipse vertical" style="width: 502px; height: 449px;" alt="Elipse vertical" src="http://aula.quinan.cl/file.php/1/elip_ver.PNG" border="0" hspace="0" /></font></em></p> <p align="justify"><em><font face="Times New Roman" size="4"> En estos gráficos notamos que el centro lo podemos calcular como el punto medio de los vértices que están en la recta que une los focos (eje focal), es decir, si sus vértices son:</font></em></p> <p align="justify"><em><font face="Times New Roman" size="4"><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"»«msub»«mi»V«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfenced»«mrow»«mi»a«/mi»«mo»,«/mo»«mi»b«/mi»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»V«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mfenced»«mrow»«mi»c«/mi»«mo»,«/mo»«mi»d«/mi»«/mrow»«/mfenced»«/math»</span></span></span></span></span></span></span></font></em></p> <p align="justify"><em><font face="Times New Roman" size="4">y el punto medio se calcula como:</font></em></p> <p align="justify"><span class="nolink"><span class="nolink"></span></span></p> <p align="justify"></p> <p align="justify"></p> <p align="justify"><em><font face="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">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»p«/mi»«mi»u«/mi»«mi»n«/mi»«mi»t«/mi»«mi»o«/mi»«mo»§nbsp;«/mo»«mi»m«/mi»«mi»e«/mi»«mi»d«/mi»«mi»i«/mi»«mi»o«/mi»«mo»=«/mo»«mfenced»«mrow»«mfrac»«mrow»«mi»a«/mi»«mo»+«/mo»«mi»c«/mi»«/mrow»«mn»2«/mn»«/mfrac»«mo»,«/mo»«mfrac»«mrow»«mi»b«/mi»«mo»+«/mo»«mi»d«/mi»«/mrow»«mn»2«/mn»«/mfrac»«/mrow»«/mfenced»«/math»</span></span></span></span></span></span></span></font></font></em></p> <p align="justify"><em><font face="times new roman,times,serif"><font size="4">En este caso, los vértices son</font></font></em><em><font face="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">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#V5«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»#V6«/mi»«/math»</span></span></span></span></span></span></span></span>, luego</span></span></span></span></span></span></span></span></span></span></font></font></em></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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><font face="times new roman,times,serif" 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">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»C«/mi»«mo»=«/mo»«mfenced»«mrow»«mfrac»«mrow»«mi»#e1«/mi»«mi»#d5«/mi»«mi»#e2«/mi»«/mrow»«mn»2«/mn»«/mfrac»«mo»,«/mo»«mfrac»«mrow»«mi»#e3«/mi»«mi»#d6«/mi»«mi»#e4«/mi»«/mrow»«mn»2«/mn»«/mfrac»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«mi»#e5«/mi»«mo»,«/mo»«mi»#e6«/mi»«/mrow»«/mfenced»«/math»</span></span></span></span></span></span></span></span></span></span></span></font></em></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p align="justify"><em><font face="times new roman,times,serif"><font size="4">Ahora, para calcular las coordenadas del foco, primero debemos <span class="nolink"><span class="nolink"><span class="nolink">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">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»c«/mi»«/math»</span></span></span></span></span></span></span></span></span></span> que es la distancia del centro al foco de la elipse. Para ello, utilizamos la siguiente relación de la elipse:</span></span></span></font></font></em></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="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">«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></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>,</font></font></em></span></span></span></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif"><font size="4">donde <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»a«/mi»«/math»</span> es la distancia del centro de la elipse al vértice contenido en el eje focal y <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»b«/mi»«/math»</span> es la distancia del centro a la recta que es perpendicular al eje focal y que pasa por el centro. <br /></font></font></em></span></span></span></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif"><font size="4">Despejando <span class="nolink"><span class="nolink"><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> y reemplazando los valores de </font></font></em></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif"><font size="4"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»a«/mi»«/math»</span></font></font></em></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif"><font size="4"> y </font></font></em></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif"><font size="4"><span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»b«/mi»«/math»</span></font></font></em></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif"><font size="4"> tenemos que <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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»«msqrt»«mrow»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/mrow»«/msqrt»«mo»=«/mo»«msqrt»«mrow»«msup»«mi»#f3«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«msup»«mi»#f4«/mi»«mn»2«/mn»«/msup»«/mrow»«/msqrt»«mo»=«/mo»«msqrt»«mrow»«mi»#f1«/mi»«mo»-«/mo»«mi»#f2«/mi»«/mrow»«/msqrt»«mo»=«/mo»«mi»#c«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>.</span></span></span></span></font></font></em></span></span></span></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><em><font face="times new roman,times,serif" size="4">Como la elipse es <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> , las coordenadas de los focos vienen dadas por</font></em></span></span></span></span></span></span></span></p> <p align="justify"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><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"»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»#F01«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F02«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></p> <p align="justify" 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"><span class="nolink"><span class="nolink"><span class="nolink">Reemplazando los valores de <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»h«/mi»«mo»,«/mo»«mi»k«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«/math»</span></span></span></span></span>, obtenemos que </span></span></span></span></span></span></span></font></p> <p align="justify"><span class="nolink"><span class="nolink"><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><font face="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></span></span></span></span></span></span></span></span></font></font></em></span></span></span></span></span></span></span></span></span></span></p> <p align="justify"><span class="nolink"><span class="nolink"><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><font face="times new roman,times,serif" size="4"><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»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»#F11«/mi»«mo»=«/mo»«mi»#F1«/mi»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F22«/mi»«mo»=«/mo»«mi»#F2«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></font></em></span></span></span></span></span></span></span></span></span></span></p> <p align="justify"><span class="nolink"><span class="nolink"><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><font face="times new roman,times,serif" size="4">Luego, la solución 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">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#sol«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span>.</font></em></span></span></span></span></span></span></span></span></span></span></p>
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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
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<p></p><span style="font-style: italic;"><font size="4"><span style="font-family: times new roman,times,serif;">Bien. Continúa de esta manera.</span></font><br /></span>
Choice 2
Answer
#op1
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<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Recuerda que la elipse es <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>, y los focos tienen coordenadas</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»#F01«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F02«/mi»«/math»</span></span></span>.</p>
Choice 3
Answer
#op2
Grade
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100%
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83.33333%
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<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Las coordenadas del centro de la elipse son</font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"> <font size="4"><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></font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Con esas coordenadas debes reemplazar la elipse en las coordenadas del foco, que son</font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4"><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»#F01«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F02«/mi»«/math»</span></span></font></p>
Choice 4
Answer
#op3
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Feedback
<p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Recuerda que la elipse tiene eje focal <span class="nolink">«math xmlns="http://www.w3.org/1998/Math/MathML"»«mi»#d3«/mi»«/math»</span> y sus 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">«math xmlns="http://www.w3.org/1998/Math/MathML"»«msub»«mi»F«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»#F01«/mi»«mo»§nbsp;«/mo»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»F«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»#F02«/mi»«/math»</span></span></font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">también las coordenadas del centro de la elipse son</font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4"><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></font></p> <p style="font-family: times new roman,times,serif; font-style: italic;"><font size="4">Con esas coordenadas debes reemplazar la elipse en las coordenadas del foco .</font></p>
Choice 5
Answer
Grade
None
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83.33333%
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70%
66.66667%
60%
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33.33333%
30%
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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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