2024 ACJC Promo Q7

The parametric equations of a curve \(C\) are

\(x=2a\cos \left( t+\frac{\pi }{6} \right)\), \(y=a\sin t\),

where \(0\le t < 2\pi \) and \(a\) is a positive constant.

A sketch of curve \(C\) is shown below, with \(C\) intersecting the positive \(x\)-axis at \(P\) and the negative \(y\)-axis at \(Q\).

1. Show that the normal to a point on the curve \(C\) with parameter \(\theta \) has gradient \(1+\sqrt{3}\tan \theta \).[3]
2. Show that \(t=\frac{4\pi }{3}\) at the point \(Q\).[1]
3. Hence, find the equation of the normal to the curve \(C\) at \(Q\).[2]

The normal to the curve \(C\) at \(Q\) intersects the \(x\)-axis at \(T\).

4. Show that the ratio \(\frac{OT}{OP}\) is independent of \(a\) and find its value exactly.[3]