If the image of the point $\mathrm{P}(1,0,3)$ in the line joining the points $\mathrm{A}(4,7,1)$ and $\mathrm{B}(3,5,3)$ is $\mathrm{Q}(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Three Dimensional Geometry

If the image of the point

P (1, 0, 3)

in the line joining the points

A (4, 7, 1)

and

B (3, 5, 3)

Q (α, β, γ)

, then

α + β + γ

is equal to

(A) $\frac{47}{3}$
(B) $\frac{46}{3}$ ✓ Correct
(C)18
(D)13

Answer: (B)

Solution

Line

A B

has direction

(3 - 4, 5 - 7, 3 - 1) = (- 1, - 2, 2)

. A general point is

R = (3 + λ, 5 + 2 λ, 3 - 2 λ)

using direction

(1, 2, - 2)

. For

R

to be the foot of the perpendicular from

P (1, 0, 3)

P R = (2 + λ, 5 + 2 λ, - 2 λ)

must be perpendicular to

(1, 2, - 2)

(2 + λ) + 2 (5 + 2 λ) + 4 λ = 0

gives

λ = - \frac{4}{3}

. Then

R = (\frac{5}{3}, \frac{7}{3}, \frac{17}{3})

, and since

R

is the midpoint of

P Q

Q = 2 R - P = (\frac{7}{3}, \frac{14}{3}, \frac{25}{3})

. Hence

α + β + γ = \frac{46}{3}

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Single correct option

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