Find the equation of the plane through the line of intersection of and perpendicular to the plane Hence find whether the plane thus obtained contains the line

OR

Find the vector and Cartesian equations of a line passing through (1, 2, -4) and perpendicular to the two lines and

Given: Planes

∴ The equation of the planes passing through the line of intersection of these planes is given by;

2x − 3y + 4z − 1 + λ(x − y + 4) = 0

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

((2 + λ), (−3 − λ), (4)) ……(i)

Perpendicular plane is

In Cartesian form (2, −1, 1) ……(ii)

According to the given condition product of (i) and (ii) should be zero.

2(2 + λ) − (−3 − λ) + 4 = 0

∴ The required equation is

The given line x − 1 = 2y − 4 = 3z − 12

(1, 2, 4) is a point on the line

∴ the plane contains the line x − 1 = 2y − 4 = 3z − 12

OR

Given: The point (1, 2, −4) and lines

The line perpendicular to these lines will be parallel to the line obtained by the cross product of these lines.

∴ The vector equation of the required line is

∴ The Cartesian equation of the required line is