NYJC/2021/JC1/Promo/P1/Q08

A curve $C$ has parametric equations
$$x=3 t^2, \quad y=a\left(t^3+1\right),$$
where $a$ is a positive constant.
(i) Sketch $C$, giving the coordinates of any point(s) where the curve meets the axes.

[2]

The tangent to $C$ at point $A(3,2 a)$ makes an angle of $\frac{\pi}{3}$ with the positive $x$-axis.
(ii) Show that $a=2 \sqrt{3}$, and find the equation of the tangent to $C$ at $A$ in the form $y=m x+c$, where $m$ and $c$ are constants to be determined.

[5]

(iii) The tangent and the normal to $C$ at $A$ meet the $x$-axis at $T$ and $N$ respectively. Find the exact area of triangle $A T N$.

[3]