Question

Let $0<\alpha <\frac{\pi }{2}$ be 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. cockwise rotation around origin through an angle $\alpha$
  
• B. anticlockwise rotation around origin through an angle $\alpha$
  
• C. reflection in the line throughorigin with slope tan $\alpha$
• D. reflection in the line through origin with slope tan $\left(\alpha /2\right)$

Answer 1

The correct option is D reflection in the line through origin with slope tan $\left(\alpha /2\right)$

Clearly $OP=OQ=1$ and $\angle QOP=\alpha -\theta -\theta =\alpha -2\theta .$





The bisector of $\angle$ QOP will be a perpendicular to PQ and also bisect it.
Hence Q is reflection of P in the line OM which makes an angle $\angle MOP+\angle POX$ with x- axis,
i.e., $\frac{1}{2}(\alpha -2\theta )+\theta =\alpha /2.$
So that slope of OM is tan $\left(\alpha /2\right)$