Final answer:
The equation for the hyperbola with the given vertices and foci is (y-0)^2/25 - (x-3)^2/64 = 1.
Step-by-step explanation:
The equation for a hyperbola with vertices at (3,5) and (3,-5) and foci at (3,9) and (3,-9) can be written in the form (y-k)^2/a^2 - (x-h)^2/b^2 = 1. The center of the hyperbola is the midpoint between the vertices, which is (3,0). The distance from the center to the vertices is a = 5, and the distance from the center to the foci is c = 9. To find b, we use the relationship c^2 = a^2 + b^2. Substituting the values, we get 81 = 25 + b^2. Solving for b, we find b = 8. Therefore, the equation of the hyperbola is (y-0)^2/25 - (x-3)^2/64 = 1.