If the shortest distance between the lines ll L₁: r=(2+λ) i+(1-3 λ) j+(3+4 λ) k, λ R L₂: r=2(1+μ) i+3(1+μ) j+(5+μ) k, μ R is m/(√(n)), where gcd(m, n)=1, then the value of m+n equals

Three Dimensional Geometry: JEE Main 2024 Mathematics question

JEE Main 2024 · 8 Apr, Morning shift · Q2538

JEE Main 20248 Apr, Morning shiftMathematicsMediumThree Dimensional Geometry249× since 2002

If the shortest distance between the lines

\begin{array}{ll} L_1: \vec{r}=(2+\lambda) \hat{i}+(1-3 \lambda) \hat{j}+(3+4 \lambda) \hat{k}, & \lambda \in \mathbb{R} \\ L_2: \vec{r}=2(1+\mu) \hat{i}+3(1+\mu) \hat{j}+(5+\mu) \hat{k}, & \mu \in \mathbb{R} \end{array}

\frac{m}{\sqrt{n}}

, where

\operatorname{gcd}(m, n)=1

, then the value of

m+n

equals