9758/2022/P1/Q03

The parametric equations of a curve are $x=\frac{1}{2}\left(\mathrm{e}^{3 t}+2 \mathrm{e}^{-3 t}\right)$ and $y=\frac{1}{2}\left(\mathrm{e}^{3 t}-2 \mathrm{e}^{-3 t}\right)$.
(a) Using calculus, find the gradient of the normal to the curve at the point where $t=\frac{1}{3} \ln 2$.

[3]

(b) By considering $x^2$ and $y^2$ or otherwise, find the cartesian equation of the curve, stating any restriction on the values of $x$.

[3]