JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Three Dimensional Geometry

Let

(α, β, γ)

be mirror image of the point

(2, 3, 5)

in the line

\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}

. Then

2 α + 3 β + 4 γ

is equal to

(A)32
(B)33✓ Correct
(C)31
(D)34

Answer: (B)

Solution

Let

R = (α, β, γ)

be the mirror image of

P (2, 3, 5)

in the line

\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}

.

The midpoint of

P R

lies on the line, and

P R

is perpendicular to the line's direction vector

(2, 3, 4)

. This perpendicularity condition alone is enough:

P R \cdot (2, 3, 4) = 0

(α - 2, β - 3, γ - 5) \cdot (2, 3, 4) = 0

2 α - 4 + 3 β - 9 + 4 γ - 20 = 0

2 α + 3 β + 4 γ = 33

JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Single correct option

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