Distance between two spheres

Question

The equations x² + y² + z² - 6z = 0 and x² - 18x + y² +16y + z² = -144 represent spheres in three dimensions. Find the distance between these two spheres. Include at least three decimal places in your answer.

Answer 1

Are you finding the distance between their centers? Or their closest points?

You can center the spheres by completing the squares:

x² + y² + (z²-6z +9)-9 = 0 or x² + y² + (z-3)² = 9 which has center (0,0,3) and radius = 3

(x² -18x + 81) - 81 +(y² +16y + 64) - 64 + z² = -144 or (x-9)² +(y+8)² +z² = 1 with center (9,-8,0) and radius = 1

Just use Pythagorean distance formula between centers.

If you want nearest points distance, just subtract the two radii from the distance between centers.