2022 EJC Promo Q3

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

\(x=at\left( t+2 \right)\), \(\qquad\) \(y=a\left( {{t}^{2}}+1 \right)\) \(\qquad\) where \(a\) is a positive constant.

1. Find, in terms of \(a\), the equation of the normal to the curve at the point where \(t=\frac{1}{2}\).[3]

Another curve \({{C}_{2}}\) has equation \({{\mathrm{e}}^{3x}}+{{\mathrm{e}}^{y}}=4\).

2. Given that the normal in part (a) is also tangent to \({{C}_{2}}\) at a point, find the exact value of \(a\).[4]