Question

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 $PQ$ is

• A. $3\sqrt{2}$
• B. 4
• C. $2\sqrt{6}$
• D. $4\sqrt{3}$

Answer 1

Answer: (C)

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