Question
Find the equation of the plane through the intersection of the planes x + y + z = 9 and 2 x + 3 y + 4 z + 5 = 0 and passing through the point (1, 1, 1).
Solution
The equation of any plane through the intersection of planes
x + y + z = 9 i.e., x + y + z – 9 = 0 and 2 x + 3 y + 4 z + 5 = 0 is
(x + y + z – 9) + k (2 x + 3 y + 4 z + 5) = 0 ...(1)
∴ it passes through the point (1, 1, 1)
∴ (1 + 1 + 1 – 9) + k (2 + 3 + 4 + 5) = 0
Putting this value of k in (1), we get,
or 
or 
or 
which is required equation of plane.
