+1072 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?
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.
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.
+118667 \(\frac{x^2}{36} - \frac{y^2}{16} = 1\\ a=6\\ b=4\)
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.
