A sphere consists of all the points that are 5 units from its center

Question

(2, 3,−6). Write an equation that describes this sphere. Does the sphere intersect the xy-plane? Explain.

Answer 1

(x-2)^2 + (y-3)^2 + (z+6)^2 = 25
is the equation. That is the general equation for a sphere with the center coordinates and radius prescribed.

That should be obvious.

If it intersected the xy plane, there would be a solution for z = 0. That would require that
(x-2)^2 + (y-3)^2 + 36 = 25
which is impossible for real x and y.

You would be have a sphere of radius 5 with its center 6 units below the plane.