Question

If $0<\alpha <\frac{\pi }{8}$ be a fixed angle. If $P(\cos \theta ,\sin \theta )$ and $Q\left(\cos \right(4\alpha -\theta ),\sin (4\alpha -\theta \left)\right)$ where $\theta ϵ(\pi ,\frac{3\pi }{2})$, then $P$ is obtained from $Q$ by

• A. Anticlockwise rotation around origin through an angle $2(\theta +\pi 2\alpha )$.
• B. Anticlockwise rotation around origin through an angle $2(\theta -\pi -2\alpha )$.
• C. Reflection in the line through origin with slope $\tan \left(\alpha \right)$.
• D. Reflection in the line through origin with slope $\tan$ $\left(2\alpha \right)$.

Answer 1

Answer: (B), (D)

The correct options are
B Anticlockwise rotation around origin through an angle $2(\theta -\pi -2\alpha )$.
D Reflection in the line through origin with slope $\tan$ $\left(2\alpha \right)$.

$\angle BOP=\theta -\pi$
$\angle BOQ=\theta -\pi -4\alpha$
So, $\angle QOP=2\theta -2\pi -4\alpha$
i.e. $P$ is obtained from $Q$ by anticlockwise rotation through an angle $(2\theta -2\pi -4\alpha )$.
If plane mirror is rotated through an angle $\alpha$ then reflected ray is
rotated through an angle ($2\alpha$).
Reflection in the line through origin with slope $\tan 2\alpha$.
Hence, options 'B' and 'D' are correct.