If the angle between the line x+11=y-12=z-22 and the plane 2x-y+ z+4=0 is such that =13, t…

Question

If the angle $	heta$ between the line $\dfrac{x+1}1=\dfrac{y-1}2=\dfrac{z-2}2$ and the plane $2x-y+\sqrt\lambda z+4=0$ is such that $\sin	heta=\dfrac13$, then the value of $\lambda$ is

Accepted Answer

$$\sin\theta=\frac{|1(2)+2(-1)+2\sqrt\lambda|}{\sqrt9\sqrt{5+\lambda}}=\frac{2\sqrt\lambda}{3\sqrt{5+\lambda}}=\frac13\implies 4\lambda=5+\lambda\implies\lambda=\frac53.$$

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