Question
Find the equation of the plane through the intersection of the planes 3x – y + 2 z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).
Solution
Any plane passing through the intersection of planes
3x – y + 2 z – 4 = 0 and x + y + z – 2 = 0 is
(3x – y + 2 z – 4) + k (x + y + z – 2) = 0 ...(1)
∴ it passes through (2, 2, 1)
∴ (6 – 2 + 2 – 4) + k (2 + 2 + 1 – 2) = 0
Putting 
or 3 (3x – y + 2 z – 4) – 2 (x + y + z – 2) = 0
or 9x – 3 y + 6 z – 12 – 2x – 2 y – 2 z + 4 = 0
or 7x – 5 y + 4 z = 8, which is required equation of plane.
