Question

Let $0<\alpha <\frac{\pi }{4}$ be a fixed angle. If $P=(\cos \theta ,\sin \theta )$ and $Q=\left(\cos \right(\alpha -\theta ),\sin (\alpha -\theta \left)\right)$ then $Q$ is obtained from $P$ by :

• A. clockwise rotation around the origin through an angle $\alpha$
• B. anticlockwise rotation around the 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

The correct option is D reflection in the line through origin with slope $\tan \frac{\alpha }{2}$

$P=(\cos \theta ,\sin \theta )$ ; $Q=(\cos (\alpha -\theta ),\sin (\alpha -\theta ))$

Angle between $P$ and $Q$ is ${\tan }^{-1}$ $\left(\frac{\sin (\alpha -\theta )-\sin \alpha }{\cos (\alpha -\theta )-\cos \alpha }\right)$

$={\tan }^{-1}\left(\tan \alpha \right)=\alpha$

$\therefore$ mid point of $P$ and $Q$ is $(\frac{\cos \theta +\cos (\alpha -\theta )}{2},\frac{\sin \theta +\sin (\alpha -\theta )}{2})$

$=(\cos \frac{\alpha }{2}\cos \left(\frac{\theta -\alpha }{2}\right),\sin \frac{\alpha }{2}\cos \left(\frac{\theta -\alpha }{2}\right))$

$=\cos \left(\frac{\theta -\alpha }{2}\right)(\cos \frac{\alpha }{2},\sin \frac{\alpha }{2})$

Which is a point on line with slope $y=\tan \frac{\alpha }{2}$

$\therefore$ $Q$ is obtained by reflection of origin with slope $\tan$ $\frac{\alpha }{2}$.

Hence, option 'D' is correct.