If the plane $2 x+y-5 z=0$ is rotated about its line of intersection with the plane $3 x-y+4 z-7=0$ by an angle of $\frac{\pi}{2}$, then the plane after the rotation passes through the point :

JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Three Dimensional Geometry

If the plane

2 x + y - 5 z = 0

is rotated about its line of intersection with the plane

3 x - y + 4 z - 7 = 0

by an angle of

\frac{π}{2}

, then the plane after the rotation passes through the point :

Answer: (C)

The family of planes through the line of intersection is

(2 x + y - 5 z) + λ (3 x - y + 4 z - 7) = 0

, which expands to

(2 + 3 λ) x + (1 - λ) y + (- 5 + 4 λ) z - 7 λ = 0

.

For perpendicularity (rotation by

π /2

), the dot product of normal vectors must be zero:

2 (2 + 3 λ) + (1 - λ) - 5 (- 5 + 4 λ) = 0

4 + 6 λ + 1 - λ + 25 - 20 λ = 0

30 = 15 λ

, so

λ = 2

.

The rotated plane is

8 x - y + 3 z - 14 = 0

. Checking option (C):

8 (1) - 0 + 3 (2) - 14 = 0

✓

Answer: (C)

JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Single correct option