Sphere touching given sphere at a point

[Read as: find the equation of the sphere which touches the given sphere at (1, 2, −2) and passes through (1, −1, 0).]

Any sphere touching S₀ at a point has the tangent plane T of S₀ at that point as the common tangent plane, so the required sphere is S₀ + kT = 0.

S₀ = x² + y² + z² + 2x − 6y + 1 = 0.

Check the point is on S₀

1 + 4 + 4 + 2 − 12 + 1 = 0, so (1, 2, −2) lies on S₀.

Tangent plane at (1, 2, −2)

Use xx₁ + yy₁ + zz₁ + (x + x₁) − 3(y + y₁) + 1 = 0:

x + 2y − 2z + (x + 1) − 3(y + 2) + 1 = 0
2x − y − 2z − 4 = 0

Sphere family

x² + y² + z² + 2x − 6y + 1 + k(2x − y − 2z − 4) = 0

Pass through (1, −1, 0)

1 + 1 + 0 + 2 + 6 + 1 + k(2 + 1 − 0 − 4) = 0
11 − k = 0   →   k = 11

Required sphere

x² + y² + z² + 2x − 6y + 1 + 11(2x − y − 2z − 4) = 0

Answer: x² + y² + z² + 24x − 17y − 22z − 43 = 0.

Check: at (1, −1, 0): 1 + 1 + 24 + 17 − 43 = 0. At (1, 2, −2): 1 + 4 + 4 + 24 − 34 + 44 − 43 = 0.