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QJEEM25Ma8c75: JEE Main 2025 Mathematics - Class 12 JEEM25Ma8c75 Solution: If the image of the point in the line is , then is equal to

If the image of the point (4,4,3)(4,4,3) in the line x−12=y−21=z−13\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3} is (α,β,γ)(\alpha, \beta, \gamma) , then α+β+γ\alpha+\beta+\gamma is equal to

Multiple Choice Options

  • A 12
  • B 9
  • C 7
  • D 8

Step-by-Step Verified Solution

JEE Main 2025 (Online) 28th January Morning Shift Mathematics - 3D Geometry Question 2 English Explanation

amp;amp;PQ→⊥(2i^+j^+3k^)amp;⇒2(2λ−3)+1(λ−2)+3(3λ−2)=0amp;⇒14λ−14=0,λ=1amp; So, Q(3,3,4)amp; Let image in R(α,β,γ)amp;amp;α+42=3,β+42=3,γ+32=4amp;(α,β,γ)=(2,2,5)amp;⇒α+β+γ=9\begin{aligned}&\begin{aligned}& \overrightarrow{\mathrm{PQ}} \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 \\& \text { So, } \mathrm{Q}(3,3,4)\end{aligned}\\&\text { Let image in } R(\alpha, \beta, \gamma)\\&\begin{aligned}& \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\end{aligned}\end{aligned}

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