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Let 0<\alpha<\frac{\pi}{2} be a fixed angle. If P=(\cos \theta, \sin \theta) and Q=(\cos (\alpha-\theta), \sin (\alpha-\theta)) then Q is obtained from P by 

 

Option: 1

clockwise rotation around origin through an angle \alpha
 


Option: 2

anti-clockwise rotation around origin through an angle \alpha
 


Option: 3

reflection in the line through origin with slope \tan \alpha
 


Option: 4

reflection in the line through origin with slope \tan \alpha / 2


Answers (1)

best_answer

See the diagram. Since there is no condition on \theta, Q can be placed either at Q_1 \text{ Or } Q_2 for a particular position of P. So option (a) and (b) cannot be definitely true.

Consider a line through origin y=m x. If Q and P are reflection of each other with line mirror y=mx

\begin{gathered} \text { (Slope of } P Q) \times m=-1 \\ \\m\left(\frac{\sin \theta-\sin (\alpha-\theta)}{\cos \theta-\cos (\alpha-\theta)}\right)=-1 \\ \\m\left(\frac{2 \cos \left(\frac{\alpha}{2}\right) \cdot \sin \left(\frac{2 \theta-\alpha}{2}\right)}{2 \sin \left(\frac{\alpha}{2}\right) \cdot \sin \left(\frac{\alpha-2 \theta}{2}\right)}\right)=-1 \end{gathered}

\begin{aligned} m\left(-\cot \frac{\alpha}{2}\right) & =-1 \\ \\m =\tan (\alpha / 2) \end{aligned}

Posted by

Pankaj Sanodiya

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