Let the lines and intersect at the point R. Let P and Q be the points lying on lines and respectively, such that and . If the point P lies in the first octant, then is equal to [2026]

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Solution

Q.
Let the lines $L_{1} : \vec{r} = \hat{i} + 2 \hat{j} + 3 \hat{k} + λ (2 \hat{i} + 3 \hat{j} + 4 \hat{k}), λ \in ℝ$ and $L_{2} : \vec{r} = (4 \hat{i} + \hat{j}) + μ (5 \hat{i} + 2 \hat{j} + \hat{k}), μ \in ℝ,$ intersect at the point R. Let P and Q be the points lying on lines $L_{1}$ and $L_{2}$ respectively, such that $| \vec{P R} |$ $= \sqrt{29}$ and $| \vec{P Q} |$ $= \sqrt{\frac{47}{3}}$ . If the point P lies in the first octant, then $27 {(Q R)}^{2}$ is equal to [2026]

1 320

2 340

3 360

4 348

Ans.
(3)

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