JEE Main 2024 — 8 April, Shift 1 — Question 9

If the shortest distance between the lines

$\mathrm{L}_{1}: \overrightarrow{\mathrm{r}}=(2+\lambda) \hat{\mathrm{i}}+(1-3 \lambda) \hat{\mathrm{j}}+(3+4 \lambda) \hat{\mathrm{k}}, \lambda \in \mathbb{R}$

$\mathrm{L}_{2}: \overrightarrow{\mathrm{r}}=2(1+\mu) \hat{\mathrm{i}}+3(1+\mu) \hat{\mathrm{j}}+(5+\mu) \hat{\mathrm{k}}, \mu \in \mathbb{R}$ is $\frac{\mathrm{m}}{\sqrt{\mathrm{n}}}$,

where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then the value of $\mathrm{m}+\mathrm{n}$ equals.