Let $(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{P}(2,3,5)$ in the plane $2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}=6$. Then $\alpha+\beta+\gamma$ is equal to

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Three Dimensional Geometry

Let

(α, β, γ)

be the image of the point

P (2, 3, 5)

in the plane

2 x + y - 3 z = 6

. Then

α + β + γ

is equal to

(A)10✓ Correct
(B)5
(C)12
(D)9

Answer: (A)

Solution

The image of point

P (2, 3, 5)

in the plane

2 x + y - 3 z = 6

is found using the formula for reflection. The perpendicular from

P

to the plane has direction ratios

(2, 1, - 3)

. Using the reflection formula:

\frac{α - 2}{2} = \frac{β - 3}{1} = \frac{γ - 5}{- 3} = - 2 (\frac{2 ( 2 ) + 3 - 3 ( 5 ) - 6}{2 ^{2} + 1 ^{2} + ( - 3 ) ^{2}}) = - 2 (\frac{4 + 3 - 15 - 6}{14}) = - 2 (\frac{- 14}{14}) = 2

. Thus

α = 2 + 2 (2) = 6

β = 3 + 2 (1) = 5

γ = 5 + 2 (- 3) = - 1

. Therefore

α + β + γ = 6 + 5 - 1 = 10

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option

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