2024 EJC Promo Q9

A curve \(C\) has parametric equations

\[x = t^2 - 1, \quad y = \frac{2}{3}t^3 - 2at \quad \text{for}\hspace{0.5em} t \in \mathbb{R}\]

where \(a\) is a positive constant. \(P\) is a point on \(C\) where \(t = a\).

1. Find the equation of the tangent to the curve at \(P\).[3]
2. \(Q\) is another point on \(C\). The tangent to the curve at \(Q\) is parallel to the tangent to the curve at \(P\). Find, in terms of \(a\), the coordinates of \(Q\).[4]
3. Find the range of values of \(a\) for which \(C\) intersects the positive \(x\)-axis.[2]