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avatar+749 

A point $P$ lies on the hyperbola\[ \frac{x^2}{36} - \frac{y^2}{16} = 1.\]The distance from $P$ to one focus is 7. What is the distance from $P$ to the other focus?

 Feb 15, 2020
 #1
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0

The foci are at (-5*sqrt(2)) and (5*sqrt(2)), and the hyperbola represents the set of points such that |PF_1 - PF_2| = 10.  So if the distance from P to one focus is 7, then the other distance is 17.

 Feb 15, 2020
 #2
avatar+749 
+2

Sorry, but that answer is incorrect, I thought it was 17 too.

 Feb 15, 2020
 #3
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For this parabola, a = 6 and b = 4, so c = sqrt(36 + 16) = 2*sqrt(13).  All points on the parabola satisfy \(|PF_1 - PF_2| = 4 \sqrt{13}\)

 

So PF_1 = 4*sqrt(13) + 7.

 Feb 16, 2020
 #4
avatar+749 
+1

I also tried 4sqrt(13)+7 but it was still incorrect.

 Feb 16, 2020
 #5
avatar+108776 
+2

\(\frac{x^2}{36} - \frac{y^2}{16} = 1\\ a=6\\ b=4\)

 

Guest above had the right idea but he used the wrong value.

 

\(PF_1=7\\ a=6\)

 

so

 

\(||PF_2|-|PF_1||=2a\\ ||PF_2|-7|=12\\ |PF_2|-7=\pm12\\ |PF_2|=-5,\quad or \quad 19\\ \text{can't be negative}\\ |PF_2|=19\\ \)

 

 

Here is the diagram.

 

 Feb 17, 2020

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