Sphere with circle as great circle

A circle is a great circle of a sphere when the sphere's centre lies on the plane of the circle.

The circle is S = x² + y² + z² − 3x + 4y − 2z − 5 = 0, P = 5x − 2y + 4z + 7 = 0.

Family of spheres

S + kP = 0:

x² + y² + z² + (5k − 3)x + (4 − 2k)y + (4k − 2)z + 7k − 5 = 0.

Centre = ( (3 − 5k)/2, (2k − 4)/2, (2 − 4k)/2 ).

Centre on the plane P = 0

5(3−5k)/2 − 2(2k−4)/2 + 4(2−4k)/2 + 7 = 0
(15 − 25k) − (4k − 8) + (8 − 16k) + 14 = 0
45 − 45k = 0   →   k = 1

(Check: 15 − 25k + 8 − 4k + 8 − 16k + 14 = 45 − 45k.)

Required sphere

Put k = 1:

x² + y² + z² + 2x + 2y + 2z + 2 = 0

Its centre is (−1, −1, −1) and 5(−1) − 2(−1) + 4(−1) + 7 = 0, so the centre is on the plane.

Answer: x² + y² + z² + 2x + 2y + 2z + 2 = 0 (centre (−1, −1, −1), radius 1).