Let 0 2 be a fixed angle. If P = ( cos , sin ) and Q = ( cos , sin ) , then Q is obtained from P by [2002]

Question

Let $0 < α < \frac{π}{2}$ be a fixed angle. If $P = (\cos θ, \sin θ) and Q = (\cos (α - θ), \sin (α - θ)),$ then $Q$ is obtained from $P$ by [2002]

Smart Achievers · Accepted Answer

(4)

$Clearly O P = O Q = 1 and ∠ Q O P = α - θ - θ = α - 2 θ$

The bisector of $∠ Q O P$ will be a perpendicular bisector of $P Q$ also. Hence $Q$ is reflection of $P$ in the line $O M$ which makes an angle $∠ M O P + ∠ P O X$ with $x$ -axis, i.e., $\frac{1}{2} (α - 2 θ) + θ = \frac{α}{2}$

So that slope of $O M$ is $\tan \frac{α}{2}$

Solution

Q.
Let $0 < α < \frac{π}{2}$ be a fixed angle. If $P = (\cos θ, \sin θ) and Q = (\cos (α - θ), \sin (α - θ)),$ then $Q$ is obtained from $P$ by [2002]

1 clockwise rotation around origin through an angle $α$

2 anticlockwise rotation around origin through an angle $α$

3 reflection in the line through origin with slope $\tan α$

4 reflection in the line through origin with slope $\tan (α / 2)$

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Solution

Q. Let 0<α<π2 be a fixed angle. If P=(cosθ,sinθ) and Q=(cos(α-θ),sin(α-θ)), then Q is obtained from P by [2002]

1 clockwise rotation around origin through an angle α

2 anticlockwise rotation around origin through an angle α

3 reflection in the line through origin with slope tanα

4 reflection in the line through origin with slope tan(α/2)