Complex Numbers 4 Question 5
5. 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
(2002, 2M)
(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 $\tan \frac{\alpha}{2}$
Show Answer
Answer:
Correct Answer: 5. (d)
Solution:
- 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 $\alpha / 2$.
