Question

Find the constant difference for a hyperbola with foci at F1 (0, -17) and F2 (0, 17), and point on the hyperbola (0, -15

Answer 1

Any point on a hyperbola has two distances to worry about, the distance to F1 and the distance to F2. All the points, when you take these two values, the difference between them will be the same - the constant difference.

You've got one point on the hyperbola, (0,-15), so that will be enough to find the constant difference.
The distance to F1 is \[\sqrt {(0-0)^2 + (-15-(-17))^2} = \sqrt{0+2^2}=\sqrt{4}=2\]
The distance to F2 is \[\sqrt{(0-0)^2 + (-15-17)^2}=\sqrt{0+ 32^2}=\sqrt{32^2}=32\]

The difference between these two distances is the constant difference: 32-2=30.