JEE Main 2025 Jan 28 Shift 1, Mathematics Q12: Lines in Space

JEE Main2025Jan 28, Shift 1Mathematics

If the image of the point $(4, 4, 3)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to

Solution

Let $P = (4, 4, 3)$ . Any point on the line is $Q (λ) = (1 + 2 λ, 2 + λ, 1 + 3 λ)$ with direction vector $d = ⟨ 2, 1, 3 ⟩$ .

The foot of perpendicular from $P$ satisfies $P Q \cdot d = 0$ :

$P Q = ⟨ 2 λ - 3, λ - 2, 3 λ - 2 ⟩$ .

$2 (2 λ - 3) + (λ - 2) + 3 (3 λ - 2) = 0 \Rightarrow 14 λ - 14 = 0 \Rightarrow λ = 1$ .

Foot $Q = (3, 3, 4)$ . The image $R = (α, β, γ)$ satisfies $Q = \frac{P + R}{2}$ , so $R = 2 Q - P = (6 - 4, 6 - 4, 8 - 3) = (2, 2, 5)$ .

$∴ α + β + γ = 2 + 2 + 5 = 9$ .

Concept behind this question

Concept-wise practice with instant solutions on Fundamenthol.