[shahkb4]

Let the line $\mathrm{L}_{1}$ be parallel to the vector $-3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and pass through the point (2,6,7) and the line $\mathrm{L}_{2}$ be parallel to the vector $2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and pass through the point (4,3,5). If the line $\mathrm{L}_{3}$ is parallel to the vector $-3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+16 \mathrm{k}$ and intersects the lines $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ at the points C and D , respectively, then $|\overrightarrow{\mathrm{CD}}|^{2}$ is equal to
[JEE MAIN 21 Jan 2026 S II]
(1) 171
(2) 290
(3) 312
(4) 89

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[shahkb4]

Solution

line $\mathrm{L}_{1} \frac{\mathrm{x}-2}{-3}=\frac{\mathrm{y}-6}{2}=\frac{\mathrm{z}-7}{4}=\lambda_{1}$
line $\mathrm{L}_{2} \frac{\mathrm{x}-4}{2}=\frac{\mathrm{y}-3}{1}=\frac{\mathrm{z}-5}{3}=\lambda_{2}$
Point $\mathrm{C}\left(-3 \lambda_{1}+2,2 \lambda_{1}+6,4 \lambda_{1}+7\right)$ on line $\mathrm{L}_{1}$
Point $\mathrm{D}\left(2 \lambda_{2}+4, \lambda_{2}+3,3 \lambda_{2}+5\right)$ on line $\mathrm{L}_{2}$

\[ \begin{aligned} & \mathrm{CD} \xrightarrow{\mathrm{drs}} 2 \lambda_{2}+3 \lambda_{1}+2, \lambda_{2}-2 \lambda_{1}-3,3 \lambda_{2}-4 \lambda_{1}-2 \\ & \text { or }-3,5,16 \\ & \frac{2 \lambda_{2}+3 \lambda_{1}+2}{-3}=\frac{\lambda_{2}-2 \lambda_{1}-3}{5}=\frac{3 \lambda_{2}-4 \lambda_{1}-2}{16} \end{aligned} \]

From first \& middle

\[ \begin{aligned} & 10 \lambda_{2}+15 \lambda_{1}+10=-3 \lambda_{2}+6 \lambda_{1}+9 \\ & \Rightarrow 13 \lambda_{2}+9 \lambda_{1}+1=0 \end{aligned} \]

From middle \& third

\[ \begin{aligned} & 16 \lambda_{2}-32 \lambda_{1}-48=15 \lambda_{2}-20 \lambda_{1}-10 \\ & \Rightarrow \lambda_{2}=12 \lambda_{1}+38 \\ & 13\left(12 \lambda_{1}+38\right)+9 \lambda_{1}+1=0 \\ & \Rightarrow 165 \lambda_{1}+495=0 \Rightarrow \lambda_{1}=-3 \\ & \Rightarrow \lambda_{2}=-36+38 \Rightarrow \lambda_{2}=2 \text { Ans. } \\ & \lambda_{1}=-3, \lambda_{2}=2 \end{aligned} \]

we get $\mathrm{C}(11,0,-5) \mathrm{D}(8,5,11)$

\[ \mathrm{CD}^{2}=9+25+256=290 \text { Answer }(2) \]

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