Correct option is (3) \(\frac{3 \sqrt{14}}{7}\)
\(\mathrm{L}_{1} \equiv \frac{\mathrm{x}-2}{1}=\frac{\mathrm{y}-4}{5}=\frac{\mathrm{z}-2}{1}=\lambda\)
\(\mathrm{P}(\lambda+2,5 \lambda+4, \lambda+2)\)
\(\mathrm{L}_{2} \equiv \frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-3}{2}\)
\(\mathrm{P}(2 \mu+3,3 \mu+2,2 \mu+3)\)
\(\lambda+2=2 \mu+3\)
\(\lambda=2 \mu+1\)
\(3 \mu+2=5 \lambda+4\)
\(3 \mu=5 \lambda+2\)
\(3 \mu=5(2 \mu+1)+2\)
\(3 \mu=10 \mu+7\)
\(\mu=-1 , \lambda=-1\)
Both satisfies (P)
\(\mathrm{P}(1,-1,1)\)
\(\mathrm{L}_{3} \equiv \frac{\mathrm{x}}{1 / 4}=\frac{\mathrm{y}}{1 / 2}=\frac{\mathrm{z}}{1}\)
\(\mathrm{L}_{3}=\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}}{4}=\mathrm{k}\)
Coordinates of \(\mathrm{Q}(\mathrm{k}, 2 \mathrm{k}, 4 \mathrm{k})\)
DR's of \(\mathrm{PQ}=<\mathrm{k}-1,2 \mathrm{k}+1,4 \mathrm{k}-1>\)
\(\mathrm{PQ} \perp\) to \(\mathrm{L}_{3}\)
\((\mathrm{k}-1)+2(2 \mathrm{k}+1)+4(4 \mathrm{k}-1)=0\)
\(\mathrm{k}-1+4 \mathrm{k}+2+16 \mathrm{k}-4=0\)
\(\mathrm{k}=\frac{1}{7}\)
\(\mathrm{Q}\left(\frac{1}{7}, \frac{2}{7}, \frac{4}{7}\right)\)
\(\mathrm{PQ}=\sqrt{\left(1-\frac{1}{7}\right)^{2}+\left(-1-\frac{2}{7}\right)^{2}+\left(1-\frac{4}{7}\right)^{2}}\)
\(=\sqrt{\frac{36}{49}+\frac{81}{49}+\frac{9}{49}}\)
\(=\frac{\sqrt{126}}{7}\)
\(\mathrm{PQ}=\frac{3 \sqrt{14}}{7}\)
