+0

+5
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+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
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
+749
+2

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

Feb 15, 2020
#3
+1

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
+749
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I also tried 4sqrt(13)+7 but it was still incorrect.

Feb 16, 2020
#5
+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