If the angle between the line x=y−12=z−3λ and the plane x+2y+3z=4 is cos−1514 then λ=

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If the angle between the line $x = \frac{y - 1}{2} = \frac{z - 3}{λ}$ and the plane $x + 2 y + 3 z = 4$ is $\cos^{- 1} (\sqrt{\frac{5}{14}})$ then $λ =$

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$\frac{5}{3}$

$\frac{2}{3}$

$\frac{1}{3}$

$\frac{3}{2}$

answer is B.

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Detailed Solution

$\sin θ = \frac{3}{\sqrt{14}} \Rightarrow \frac{1 + 4 + 3 λ}{\sqrt{5 + λ^{2}} \sqrt{14}} \Rightarrow {(5 + 3 λ)}^{2} = 9 (5 + λ^{2}) \Rightarrow 25 + 9 λ^{2} + 30 λ = 45 + 9 λ^{2} \Rightarrow λ = \frac{2}{3}$