# Distance from sphere to plane

edited January 17 in Problems

Find. the distance from the sphere x^2+y^2+z^2 - 6z = -4 to the xy-plane.

• edited January 17

To find where this sphere is centered we can start by completing the square on the z terms. (This is assuming that the -6x was meant to be a -6z)
x^2 +y^2 + z^2 -6z + 9 -9= -4

x^2 + y^2 + (z-3)^2 -9= -4

Adding the constant term over we find that this is a sphere centered at (0,0,3) with a radius of sqrt(5).
x^2 + y^2 + (z-3)^2 = 5

The point on the sphere closest to the xy-plane can be found by adding -sqrt(5) to the z value of the sphere's center point. Therefore the sphere is (3-sqrt(5)) units away from the xy-plane.