2020 NJC P2 Q4

The curve \({{C}_{1}}\) has parametric equations

\(x=4+4\sin \theta \), \(y=2\cos \theta \),

where \(-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}\).

Sketch \({{C}_{1}}\), labelling any points where \({{C}_{1}}\) crosses the axes.[2]

The curve \({{C}_{2}}\) has equation \(y={{x}^{2}}-ax\), where \(a\) is a positive constant.

1. In the case where \(a=\frac{7}{2}\), it is given that \(\left( 4,2 \right)\) is a point of both \({{C}_{1}}\) and \({{C}_{2}}\).
   Sketch \({{C}_{2}}\) on the same diagram as \({{C}_{1}}\), labelling the points of intersection of \({{C}_{1}}\) and \({{C}_{2}}\).
   Hence, find the exact area of the region bounded by \({{C}_{1}}\) and \({{C}_{2}}\).[5]
2. State the range of values of \(a\) such that \({{C}_{1}}\) and \({{C}_{2}}\) meet at two distinct points.[1]