Question

Find the equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin.

Answer 1

The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 is

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

Since this plane passes through the origin, we get,

(0 + 0 + 0 + 4) + λ(0 + 0 + 0 + 1) = 0

∴ 4 + λ = 0

∴ λ = −4

Substituting λ = −4 in equation (1), we get

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

∴ x + 2y + 3z + 4 − 16x − 12y − 8z − 4 = 0

∴ −15x − 10y − 5z = 0

∴ 3x + 2y + z = 0

This is the required equation of the plane.