Question

If $0<\alpha <\frac{\pi }{2}$ is a fixed angle. If $P=(\cos \theta ,\sin \theta )$ and $Q=\{\cos (\alpha -\theta ),\sin (\alpha -\theta )\}$ then $Q$ is obtained from $P$ by

• A. clockwise rotation around origin through an angle $\alpha$
• B. anti-clockwise rotation around origin through an angle $\alpha$
• C. reflection in the line through origin with slope $\tan \alpha$
• D. reflection in the line through origin with slope $\tan \frac{\alpha }{2}$

Answer 1

Answer: (D)

In the Argand plane, $P$ is represented by $e^{i0}$ and $Q$ is represented by $e^{i(\alpha -\theta )}$ Now, rotation about a line with angle $\alpha$ is given by $e^{\theta }\to e^{(\alpha -\theta )}$. Therefore, $Q$ is obtained from $P$ by reflection in the line making an angle $\alpha /2$.