Let (alpha,beta,gamma) be the mirror image of the point (2,3,5) in the line (x-1)/(2)=(y-2)/(3)=(z-3)/(4) . Then, 2alpha+3beta+4gamma is equal to

3D Geometry: Let (alpha,beta,gamma) be the mirror image of the point (2,3,5) in… | JEE Main 2024

JEE Main 2024 (Online) 31st January Evening ShiftQuestion 1

Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Then, $2\alpha+3\beta+4\gamma$ is equal to

3D Geometry

$31$$32$$33$✓ Correct answer$34$

Solution

Foot $F=(1+2t,2+3t,3+4t)$; $(2t-1,3t-1,4t-2)\cdot(2,3,4)=0\Rightarrow 29t=13$, $t=\frac{13}{29}$. Image $=2F-(2,3,5)$. $F\cdot(2,3,4)=2+6+12+29t=20+13=33$. So $2\alpha+3\beta+4\gamma=2(33)-(4+9+20)=33$. Option (C).

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