Let $\begin{array}{l} L_{1} : \bar{r} = (2 \hat{i} - 3 \hat{j} + 7 \hat{k}) + λ (2 \hat{i} + p \hat{j} + 5 \hat{k}), λ \in R, \\ L_{2} : \bar{r} = (\hat{i} - 4 \hat{k}) + μ (3 \hat{i} - p \hat{j} + p \hat{k}), μ \in R . \end{array}$
And if they are perpendicular to each other for all values of $λ a n d μ,$ then which of the following may be true :

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The equation of the plane containing $L_{1} a n d L_{2}$ is 4x-17y-5z-24=0 for one of the possible values of “p”

p=6

$p^{2} + 7 p + 6 = 0$

$L_{1}, L_{2}$ are non –co planar lines for one of the possible values of “p”

answer is A, B, C, D.

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Detailed Solution

$\begin{array}{l} [\bar{a} - \bar{c} \bar{b} \bar{d}] = 0 \Rightarrow p = - 1 \\ p = - 1 i s r o o t o f p^{2} + 7 p + 6 = 0 \\ p = 6 \Rightarrow L_{1} a n d L_{2} a r e c o p l a n a r \\ p = - 1 \Rightarrow L_{1} a n d L_{2} a r e n o n c o p l a n a r \\ a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0 \\ (2) (3) + (p) (- p) + (5) p = 0 \\ p^{2} - 5 p - 6 = 0 \\ (p - 6) (p + 1) = 0 \\ p = 6, p = - 1 \end{array}$