Intersection of Lines and Distances (Jee Mains)

3D Line Equations and Skew

Hard

Intersection of Lines and Distances

Let the lines L₁: $\vec r = \hat i + 2\hat j + 3\hat k + \lambda \left( {2\hat i + 3\hat j + 4\hat k} \right)$ λ∈ℝ and $\vec r = \left( {4\hat i + \hat j} \right)$ + $\mu \left( {5\hat i + 2\hat j + \hat k} \right)$ μ∈ℝ intersect at the point R. Let P and Q be the points lying on lines L₁ and L₂, respectively, such that $\left| {\overrightarrow {PR} } \right| = \sqrt {29}$ and $\left| {\overrightarrow {PQ} } \right| = \sqrt {\frac{{47}}{3}}$ If the point P lies in the first octant, then 27(QR)² is equal to -

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