JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Three Dimensional Geometry

If the foot of the perpendicular drawn from

(1, 9, 7)

to the line passing through the point

(3, 2, 1)

and parallel to the planes

x + 2 y + z = 0

and

3 y - z = 3

(α, β, γ)

, then

α + β + γ

is equal to

(A)-1
(B)3
(C)1
(D)5✓ Correct

Answer: (D)

Solution

A direction of the line is perpendicular to both plane normals

(1, 2, 1)

and

(0, 3, - 1)

, hence along their cross product

(1, 2, 1) \times (0, 3, - 1) = (- 5, 1, 3)

. A general point of the line is

M = (3 - 5 t, 2 + t, 1 + 3 t)

. For

M

to be the foot of the perpendicular from

P (1, 9, 7)

, the vector

P M = (2 - 5 t, t - 7, 3 t - 6)

must be orthogonal to

(- 5, 1, 3)

- 5 (2 - 5 t) + (t - 7) + 3 (3 t - 6) = 35 t - 35 = 0

, so

t = 1

. Then

M = (- 2, 3, 4)

and

α + β + γ = - 2 + 3 + 4 = 5

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Single correct option

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