The plane $l x + m y = 0$ is rotated about its line of intersection with the XY – plane through an angle $α .$ Then the equation of the plane is $l x + m y + n z = 0,$ where $n$ is ____

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$\pm \sqrt{(l^{2} + m^{2})} \tan α$

$\pm \sqrt{(l^{2} + m^{2})} \sin α$

$\pm \sqrt{(l^{2} + m^{2})} \cot α$

$\pm \sqrt{(l^{2} + m^{2})} \cos α$

answer is C.

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Detailed Solution

Required plane is $l x + m y + λ z = 0$

$\cos α = \frac{l^{2} + m^{2} + 0}{\sqrt{l^{2} + m^{2} + λ^{2}} \sqrt{l^{2} + m^{2}}} \Rightarrow \sec^{2} α = \frac{l^{2} + m^{2} + λ^{2}}{l^{2} + m^{2}} \Rightarrow \tan^{2} α = \frac{λ^{2}}{l^{2} + m^{2}}$

$λ = \pm \sqrt{l^{2} + m^{2}} \tan α$

Required plane equation is $l x + m y \pm \sqrt{l^{2} + m^{2}} \tan α z = 0$

$∴ n = \pm \sqrt{l^{2} + m^{2}} \tan α$

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