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 $e^{i 0}$ and $Q$ is represented by $e^{i(\alpha-\theta)}$ Now, rotation about a line with angle $\alpha$ is given by $e^{\theta} \rightarrow e^{(\alpha-\theta)}$. Therefore, $Q$ is obtained from $P$ by reflection in the line making an angle $\alpha / 2$.