JEE Main · 2015 · 04 Apr 2015 · Offline · Three Dimensional Geometry

The equation of the plane containing the line

2 x - 5 y + z = 3; x + y + 4 z = 5

, and parallel to the plane,

x + 3 y + 6 z = 1

, is :

Answer: (B)

Solution

Every plane through the given line belongs to the pencil

(2 x - 5 y + z - 3) + λ (x + y + 4 z - 5) = 0

, whose normal is

(2 + λ, - 5 + λ, 1 + 4 λ)

. For it to be parallel to

x + 3 y + 6 z = 1

this normal must be proportional to

(1, 3, 6)

; matching the first two components,

3 (2 + λ) = - 5 + λ

gives

λ = - \frac{11}{2}

. Substituting and clearing fractions, the plane becomes

x + 3 y + 6 z = 7

-- the member of the pencil with the right tilt but offset from the parallel plane.

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option