Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

If $\theta$ is the angle between line and plane then $\left(\frac{\pi }{2}-\theta \right)$ is the angle between line and normal to plane given by $$\begin{array}{rl}& \cos \left(\frac{\pi }{2}-\theta \right)=\frac{(\hat{i}+2\hat{j}+2\hat{k})\cdot (2\hat{i}-\hat{j}+\sqrt{\lambda }\hat{k})}{3\sqrt{4+1+\lambda }}\\ & \cos \left(\frac{\pi }{2}-\theta \right)=\frac{2-2+2\sqrt{\lambda }}{3\times \sqrt{5}+\lambda }\\ & \Rightarrow \sin \theta =\frac{2\sqrt{\lambda }}{3\sqrt{5}+\lambda }=\frac{1}{3}\Rightarrow 4\lambda =5+\lambda \Rightarrow \lambda =\frac{5}{3}.\end{array}$$