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

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$2 \sqrt{6}$

$4 \sqrt{3}$

$3 \sqrt{2}$

answer is C.

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Detailed Solution

$\begin{array}{l} P (2 λ + 1, λ + 3, 2 λ + 2) \\ Q (μ + 2, 2 μ + 2, 3 μ + 3) \\ \frac{2 λ - μ - 1}{1} = \frac{λ - 2 μ + 1}{- 1} = \frac{2 λ - 3 μ - 1}{- 2} \\ \Rightarrow λ = μ = 3 P \equiv (7, 6, 8) and Q (5, 8, 12) \\ \Rightarrow PQ = \sqrt{2^{2} + 2^{2} + 4^{2}} = \sqrt{24} = 2 \sqrt{6} \end{array}$