Let $$0$$<α<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 the

Correct option is 4reflection in the line through the origin with slope, tan α2

Questions

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

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clockwise rotation around the origin through an angle α

anticlockwise rotation around the origin through an angle α

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

reflection in the line through the origin with slope, tan α2

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detailed solution

Correct option is D

Clearly, points P(cos⁡θ,sin⁡θ) and Q(cos⁡(α−θ)sin⁡(α−θ) ) lie on circle of unit radius. In the figure, ∠POX=θ and ∠QOX=α−θ . ∴ ∠QOP=α−2θ Now, ΔQOP is isosceles. Therefore, altitude or angle bisector OM is perpendicular bisector of PQ. ∠MOP=α−2θ2=α2−θ∠MOX=θ+α2−θ=α2Thus, point P is reflection of point Q inline OM having slope, tan α2

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