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

Mathematics · 3D Geometry

JEE Main 2025 — 28 January, Morning Shift — Question 12

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 $(\alpha, \beta, \gamma)$ , then $\alpha+\beta+\gamma$ is equal to

Option A:
9
Correct
Option B:
12
Option C:
8
Option D:
7

Answer: A

Step-by-step solution

$\overrightarrow{\mathrm{P} \mathrm{Q}} \perp(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}})$

$\Rightarrow 2(2 \lambda-3)+1(\lambda-2)+3(3 \lambda-2)=0$

$\Rightarrow 14 \lambda-14=0, \lambda=1$

So, $\mathrm{Q}(3,3,4)$

Let image in $\mathrm{R}(\alpha, \beta, \gamma)$

$\frac{\alpha+4}{2}=3, \frac{\beta+4}{2}=3, \frac{\gamma+3}{2}=4$

$(\alpha, \beta, \gamma)=(2,2,5)$

$\Rightarrow \alpha+\beta+\gamma=9$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 28 January, Morning Shift
Subject: Mathematics
Chapter: 3D Geometry
Topic: Direction Cosines and Direction Ratios