The equation x^2 + y^2 + z^2 - 4x + 6z + a = 0 defines a sphere of radius 4. Find a.

Answer 1

x^2 + y^2 + z^2 - 4x + 6z + a = 0
x^2 - 4x + .... + y^2 + z^2 + 6z + .... = -a
(x^2 - 4x + 4) + y^2 + (x^2 + 6z + 9) = -a+4+9
(x-2)^2 + y^2 + (z+3)^2 = 13-a

comparing this with the standard form:
r^2 = 13-a = 4
a = 9

Answer 2

if the radius is 4, then r^2 = 16