Straight Line Question 209

Question: Let $ 0<\alpha <\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 the

Options:

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 the origin with slope $ \tan \alpha $

D) Reflection in the line through the origin with slop $ \tan (\alpha /2) $

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Answer:

Correct Answer: D

Solution:

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