Centre and radius of circle of sphere section

Sphere: x² + y² + z² − 8x + 4y + 8z − 45 = 0. Comparing with x² + y² + z² + 2ux + 2vy + 2wz + d = 0: u = −4, v = 2, w = 4, d = −45.

Centre C = (−u, −v, −w) = (4, −2, −4)
Radius R = √(16 + 4 + 16 + 45) = √81 = 9

Plane: x − 2y + 2z = 3, normal (1, −2, 2), |n| = 3.

Distance of C from the plane

p = |4 − 2(−2) + 2(−4) − 3| / 3 = |4 + 4 − 8 − 3|/3 = 1

Radius of the circle

r = √(R² − p²) = √(81 − 1) = √80 = 4√5 ≈ 8.94

Centre of the circle

It is the foot of the perpendicular from C on the plane. The perpendicular is (x − 4)/1 = (y + 2)/(−2) = (z + 4)/2 = t, so (4 + t, −2 − 2t, −4 + 2t). Put in the plane:

(4 + t) − 2(−2 − 2t) + 2(−4 + 2t) = 3
4 + t + 4 + 4t − 8 + 4t = 3
9t = 3  →  t = 1/3

Centre = (4 + 1/3, −2 − 2/3, −4 + 2/3) = (13/3, −8/3, −10/3).

Check: 13/3 + 16/3 − 20/3 = 3 ✓.

Answer: centre (13/3, −8/3, −10/3), radius 4√5.