Sphere with given circle as great circle

Given. Circle: S: x² + y² + z² + 7y − 2z + 2 = 0 with the plane π: 2x + 3y + 4z − 8 = 0.

Sphere through the circle

Any sphere through the circle is

x² + y² + z² + 7y − 2z + 2 + λ(2x + 3y + 4z − 8) = 0

x² + y² + z² + 2λx + (7 + 3λ)y + (4λ − 2)z + (2 − 8λ) = 0

Its centre is (−λ, −(7 + 3λ)/2, (2 − 4λ)/2).

Condition for a great circle

The given circle is a great circle when the centre of the sphere lies on the plane π:

2(−λ) + 3(−(7 + 3λ)/2) + 4((2 − 4λ)/2) − 8 = 0
−2λ − (21 + 9λ)/2 + 4 − 8λ − 8 = 0

Multiply by 2: −4λ − 21 − 9λ + 8 − 16λ − 16 = 0, so −29λ − 29 = 0, λ = −1.

Equation of the sphere

x² + y² + z² + 7y − 2z + 2 − (2x + 3y + 4z − 8) = 0
x² + y² + z² − 2x + 4y − 6z + 10 = 0

Centre (1, −2, 3), radius √(1 + 4 + 9 − 10) = 2. Check: centre on the plane: 2 − 6 + 12 − 8 = 0 ✓.

Answer: x² + y² + z² − 2x + 4y − 6z + 10 = 0.