First slide

Equation of line in Straight lines

Question

$Let 0 < α < π / 2 be a fixed angle. If P \equiv (\cos θ, \sin θ) and Q \equiv (\cos (α - θ), \sin (α - θ)), then Q is obtained from P b y t h e$

Difficult

A

clockwise rotation around the origin through an angle $α$
B

anticlockwise rotation around the origin through an angle $α$
C

reflection in the line through the origin with slope, tan $α$
D

reflection in the line through the origin with slope, tan $(\frac{α}{2})$

Solution

$Clearly, points P (\cos θ, \sin θ) and Q (\cos (α - θ) \sin (α - θ)) lie on circle of unit radius.$
$In the figure, ∠ P O X = θ and ∠ Q O X = α - θ .$
$\begin{aligned} ∴ ∠ Q O P = α - 2 θ \\ Now, Δ Q O P is isosceles. \end{aligned}$
Therefore, altitude or angle bisector OM is perpendicular bisector of PQ.
$\begin{aligned} ∠ M O P = \frac{α - 2 θ}{2} = \frac{α}{2} - θ \\ ∠ M O X = θ + (\frac{α}{2} - θ) = \frac{α}{2} \end{aligned}$
Thus, point P is reflection of point Q inline OM having slope, tan $(\frac{α}{2})$

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