IIT-JEE 2008 · Paper 2, Mathematics question 5

Common form. With $\phi=\beta-\alpha$, all three points are $\big(\cos(\phi+s),\ \sin(\beta-s)\big)$: Q at $s=0$, R at $s=\theta$, and P at $s=\frac\pi2$ (since $\cos(\phi+\frac\pi2)=-\sin\phi$ and $\sin(\beta-\frac\pi2)=-\cos\beta$).

An ellipse. $x=\cos(s+\phi)$, $y=\cos\left(s+\frac\pi2-\beta\right)$ trace an ellipse unless the phase difference $\frac\pi2-\beta-\phi=\frac\pi2-2\beta+\alpha$ is a multiple of π. For $0<\alpha,\beta<\frac\pi4$ it lies strictly between 0 and π, so the curve is a genuine ellipse.

Conclusion. Three distinct points of an ellipse are never collinear.

(D)