The intersection of the sphere with the plane is a circle. What is the z-coordinate of the center of the circle?
well, there are many points on the circle, and potentially an infinite amount of z-points, meaning, all the coordinates on the circle have a different z-coordinate. the problem makes sense to me if you state a unique point, which makes sense to be the center.
if that's not what they want (did you type the question exactly as it appeared?), then simply solve for z in the last equation i gave you
2. The equation you got from Jhevon describes a sphere. This sphere is the smallest which fits exactly into the circle which contains the points of intersection between the plane and the original sphere.
3. Therefore the plane cuts the 2nd sphere at it's equator and therefore the center of the sphere is simultaneously the center of the circle you are looking for.
4. In the attached sketch I haven't drawn the lower halfsphere so you have a better "insight".