The standard form of a vertical hyperbola centered at (h,k) is of the form (y - k)^2/a^2 - (x - h)^2/b^2 = 1.
We have been given the vertices of the hyperbola (0,9) and (0,-9). Since the hyperbola is vertical and the vertices lie on the y-axis, we gather that the center of the hyperbola is at the origin (0,0). Therefore, we have h = 0 and k = 0 in our given equation.
Next, we determine the length of the semi-major axis, denoted by 'a'. This is the distance from the center of the hyperbola to either vertex. Considering the provided vertices, we find that a = 9.
At this stage, our hyperbola equation becomes (y - 0)^2/9^2 - (x - 0)^2/b^2 = 1. This simplifies to y^2/81 - x^2/b^2 = 1.
Now, we need to find the value of 'b'. This would generally be obtained by calculating the slope of the asymptotes, which is given by the ratio b/a. However, because the slope of the asymptotes has not been provided in the question, we cannot find the value for 'b'.
So, the final form of the hyperbola equation is y^2/81 - x^2/b^2 = 1, but without the slope of the asymptote, the exact value for 'b' cannot be obtained.