2022 EJC Promo Q3
A curve ${{C}_{1}}$ has parametric equations
$x=at\left( t+2 \right)$, $y=a\left( {{t}^{2}}+1 \right)$ where $a$ is a positive constant.
(a)
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 ${{\text{e}}^{3x}}+{{\text{e}}^{y}}=4$.
(b)
Given that the normal in part (a) is also tangent to ${{C}_{2}}$ at a point, find the exact value of $a$.
[4]
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