Verified Solution • 2025–26 Session moderate QJEEM24Ma2500: JEE Main 2024 Mathematics - Class 12 JEEM24Ma2500 Solution: Let a line passing through the point intersect the lines at and … Let a line passing through the point (−1,2,3)(-1,2,3)(−1,2,3) intersect the lines L1:x−13=y−22=z+1−2L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}L1:3x−1=2y−2=−2z+1 at M(α,β,γ)M(\alpha, \beta, \gamma)M(α,β,γ) and L2:x+2−3=y−2−2=z−14L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}L2:−3x+2=−2y−2=4z−1 at N(a,b,c)N(a, b, c)N(a,b,c) . Then, the value of (α+β+γ)2(a+b+c)2\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}(a+b+c)2(α+β+γ)2 equals __________. Step-by-Step Verified Solution amp;M(3λ+1,2λ+2,−2λ−1)∴α+β+γ=3λ+2amp; N(−3μ−2,−2μ+2,4μ+1)∴a+b+c=−μ+1\begin{aligned}& \mathrm{M}(3 \lambda+1,2 \lambda+2,-2 \lambda-1) \quad \therefore \alpha+\beta+\gamma=3 \lambda+2 \\& \mathrm{~N}(-3 \mu-2,-2 \mu+2,4 \mu+1) \quad \therefore \mathrm{a}+\mathrm{b}+\mathrm{c}=-\mu+1\end{aligned}amp;M(3λ+1,2λ+2,−2λ−1)∴α+β+γ=3λ+2amp; N(−3μ−2,−2μ+2,4μ+1)∴a+b+c=−μ+1 amp;3λ+2−3μ−1=2λ−2μ=−2λ−44μ−2amp;3λμ+2μ=3λμ+λamp;2μ=λamp;2λμ−λ=λμ+2μamp;λμ=λ+2μamp;⇒λμ=2λ\begin{aligned}& \frac{3 \lambda+2}{-3 \mu-1}=\frac{2 \lambda}{-2 \mu}=\frac{-2 \lambda-4}{4 \mu-2} \\& 3 \lambda \mu+2 \mu=3 \lambda \mu+\lambda \\& 2 \mu=\lambda \\& 2 \lambda \mu-\lambda=\lambda \mu+2 \mu \\& \lambda \mu=\lambda+2 \mu \\&\Rightarrow \lambda \mu=2 \lambda\end{aligned}amp;−3μ−13λ+2=−2μ2λ=4μ−2−2λ−4amp;3λμ+2μ=3λμ+λamp;2μ=λamp;2λμ−λ=λμ+2μamp;λμ=λ+2μamp;⇒λμ=2λ ⇒amp;μ=2(λ≠0)∴amp;λ=4amp;α+β+γ=14amp;a+b+c=−1amp;(α+β+γ)2(a+b+c)2=196\begin{aligned}\Rightarrow \quad & \mu=2 \quad(\lambda \neq 0) \\\therefore \quad & \lambda=4 \\& \alpha+\beta+\gamma=14 \\& a+b+c=-1 \\& \frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}=196\end{aligned}⇒∴amp;μ=2(λ=0)amp;λ=4amp;α+β+γ=14amp;a+b+c=−1amp;(a+b+c)2(α+β+γ)2=196 Related Learning Resources & Exam Preparation JEE Main PYQ Question Bank Browse year-wise authentic previous years' questions with solutions Authentic CBT Mock Tests Simulate official exam environment with live timer and percentile scoring
