Find the equation of the sphere if one of its diameters has endpoints (8,7,6) and (9,9,9).

we know that

the equation of the sphere is equal to

where

(h,k,l) is the center of the sphere

r is the radius of the sphere

Step 1

Find the center of the sphere (h,k,l)

Let

A(8,7,6)  B(9,9,9)

we know that

the midpoint of the diameter is the center point

Find the midpoint AB

in the x-coordinate

=(8+9)/2------> 8.5

in the y-coordinate

=(7+9)/2------> 8

in the z-coordinate

=(6+9)/2------> 7.5

the center is (8.5,8,7.5)

Step 2

Find the length of the diameter (distance AB)

A(8,7,6)  B(9,9,9)

the diameter is

the radius is

Step 3

Find the equation of the sphere

the center is (8.5,8,7.5)

Substitute

therefore

the answer is

The equation of the sphere is