SOLUTION: find the standard equation for the hyperbola with the given characteristics: center (1, 2) one vertex (1, 18) one focus (1, -18)

Instead of doing your exact problem, I'll do one exactly like it, given
the exact same parts.

Since the vertex (4,9) is above the center (4,-3) the hyperbola
opens up and down.  So its equation is



and since the center is (h,k) = (4,-3), we can fill in h=4 and k=-3.



From the center (4,-3) to the vertex (4,9) is 12 units, so a=12.
We fill that in:





We just need b.  To find b we need the Pythagorean relation for
all hyperbolas, which is .

We have a=12.  We can find c because c is the distance from the center
to the focus.  The distance between the center (4,-3) and the focus
(4,-16) is 13 units, so c=13.  We substitute c=13 in the Pythagorean
relation:







Answer:



Now use this as a model to do your problem, which is exactly like it.

Edwin