Find an equation of the sphere that passes through the point (7, 5, −3) and has center (5, 8, 3).

The general equation of the sphere centered at (a,b,c) with radius r is:

(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2.  

In this case we have (x-5)^2 + (y-8)^2 + (z-3) = r^2.

We know that the point (7, 5, -3) satisfies this equation.  Subbing those three coordinates in the above equation will enable us to determine the radius, r:

(7-5)^2 + (5-8)^2 + (-3-3)^2 = r^2.  Then:

  4        +     9        +   36 = r^2.    r^2 is then 49, and the radius, r, must then be +7.

In summary, the equation of the given sphere is:

(x-5)^2 + (y-8)^2 + (z-3)^2 = 7^2.