N2013 P1 Q11

A curve \(C\) has parametric equations

\(x=3{{t}^{2}}\), \(y=2{{t}^{3}}\).

1. Find the equation of the tangent to \(C\) at the point with parameter \(t\).[3]
2. Points \(P\) and \(Q\) on \(C\) have parameters \(p\) and \(q\) respectively. The tangent at \(P\) meets the tangent at \(Q\) and some point \(R\). Show that the \(x\)-coordinates of \(R\) is \({{p}^{2}}+pq+{{q}^{2}}\), and find the \(y\)-coordinates of \(R\) in terms of \(p\) and \(q\). Given that \(pq=-1\), show that \(R\) lies on the curve with equation \(x={{y}^{2}}+1\).[5]

A curve \(L\) has equation \(x={{y}^{2}}+1\). The diagram shows parts of \(C\) and \(L\) for which \(y\ge 0\). The curves \(C\) and \(L\) touch at the point \(M\).

3. Show that \(4{{t}^{6}}-3{{t}^{2}}+1=0\) at \(M\). Hence, or otherwise, find the exact coordinates of \(M\).[3]
4. Find the exact value of the area of the shaded region bounded by \(C\) and \(L\) for which \(y\ge 0\).[6]