Correct option (D) P, Q and R are non-collinear
It is known that three points (x1, y1), (x2, y2) and (x3, y3) are collinear if and only if
Therefore, from R3 - R1sinθ - R2cosθ, we have
This implies that x1y2 - x2y1 = 0 or sinθ + cosθ = 1. Since 0 < θ < π/4, sinθ + cosθ ≠ 1. Therefore, x1y2 x2y1 = 0.
This implies that
-sin(β - α)sin β + cosβcos(β - α) = 0
cos(β - α + β) = 0
Hence,2β - α = π/2 which is impossible because 0 < α and β < π/4. Therefore
2β - α ≠ π/2
Thus x1y2 - x2y1 ≠ 0. Hence, the points P, Q and R are non-collinear.