Can someone show me the steps to answering this Algebra conic sections prob? Don't understand it!?

Question

It says "Write an equation for the hyperbola that satisfies the set of conditions" and the conditions are
verticies (-4,1) and (-4,9), foci (-4, 5± SQRT 97)
Are there any websites that can show step by step how to answer this? I can't find anything that helps!

Answer 1

For a hyperbola that opens up/down, the formula is:

(y - k)^2/b^2 - (x - h)^2/a^2 = 1
where:
h,k is the center
h, k +/- c are the foci where c^2 = a^2 + b^2

So, as given above:
Vertices are at (-4,1) and (-4,9), so the center is (-4,5). h = -4, k = 5.
Focus is -4, 5 +/- √97, so c = √97, and therefore a^2 + b^2 = 97.

a^2 + b^2 = 97 can have a whole number for a and b if a and be equal 9 (9^2 = 81) and 4 (4^2 = 16) respectively. The two hyperbolae that meet your criteria then are:

(y - 5)^2/9 - (x + 4)^2/4 = 1
(y - 5)^2/4 - (x + 4)^2/9 = 1