JEE Main · 2024 · 05 Apr 2024 · Shift 2 · Three Dimensional Geometry

Let

(α, β, γ)

be the image of the point

(8, 5, 7)

in the line

\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{5}

. Then

α + β + γ

is equal to

(A)20
(B)16
(C)14✓ Correct
(D)18

Answer: (C)

Solution

Line:

\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{5} = t

, direction

(2, 3, 5)

; point

A (8, 5, 7)

.
Foot of perpendicular

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

;

A M \cdot (2, 3, 5) = 0

2 (2 t - 7) + 3 (3 t - 6) + 5 (5 t - 5) = 38 t - 57 = 0 ⟹ t = \frac{3}{2}

M = (4, \frac{7}{2}, \frac{19}{2})

Image point

A^{'} = 2 M - A = (8 - 8, 7 - 5, 19 - 7) = (0, 2, 12)

α + β + γ = 0 + 2 + 12 = 14

JEE Main · 2024 · 05 Apr 2024 · Shift 2 · Single correct option

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