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  4. Consider the lines L1 and L2 given by L1: (x-1/2)=(y-3/1)=(z-2/2) L2: (x-2/1)=(y-2/2)=(z-3/3) . A line L3 having direction ratios 1,-1,-2, intersects L1 and L2 at the points P and Q respectively. Then the length of line segment P Q is

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Q. Consider the lines $L_1$ and $L_2$ given by
$L_1: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2} $
$ L_2: \frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3} $.
A line $L_3$ having direction ratios $1,-1,-2$, intersects $L_1$ and $L_2$ at the points $P$ and $Q$ respectively. Then the length of line segment $P Q$ is

JEE MainJEE Main 2023Three Dimensional Geometry

Solution:

Let $P =(2 \lambda+1, \lambda+3,2 \lambda+2)$
Let $Q=(\mu+2,2 \mu+2,3 \mu+3)$
$ \Rightarrow \frac{2 \lambda-\mu-1}{1}=\frac{\lambda-2 \mu+1}{-1}=\frac{2 \lambda-3 \mu-1}{-2} $
$\Rightarrow \lambda=\mu=3 \Rightarrow P (7,6,8) \text { and } Q (5,8,12) $
$ PQ =2 \sqrt{6}$