2025 RI P1 Q11

A curve \(C\) has parametric equations

\[x = 2 \cos \theta + \cos 2\theta, \quad y = 2 \sin \theta - \sin 2\theta,\]

where \(0 < \theta < 2\pi\) and \(\theta \neq \frac{2\pi}{3}, \pi, \frac{4\pi}{3}\).

The point \(P\) with parameter \(p\) lies on \(C\), and the lines \(M_p\) and \(N_p\) are the tangent and normal to \(C\) at \(P\) respectively.

1. It is given that\[\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2} \quad \text{and}\] \[\cos A - \cos B = -2 \sin \frac{A+B}{2} \sin \frac{A-B}{2}.\] Using these two results, show that the gradient of \(M_p\) is \(-\tan \left( \frac{p}{2} \right)\).[2]
2. For \(p = \frac{\pi}{3}\), \(M_p\) and \(N_p\) cut the \(x\)-axis at points \(Q\) and \(R\) respectively. Find the area of \(\Delta PQR\).[4]

3. The point \(S\) with parameter \(s\) lies on \(C\), and the line \(N_s\) is the normal to \(C\) at \(S\). It is given that \(M_p\) and \(N_s\) are parallel.

   Given \(p \neq \frac{\pi}{3}, \frac{5\pi}{3}\) and \(p < s\), by using the result \(\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\), find an equation relating \(p\) and \(s\).

   [2]