NYJC/2021/JC2/Prelim/P1/Q08

A curve $C$ has parametric equations
$$x=a \sin ^5 t, \quad y=a \cos ^5 t$$
where $a$ is a positive constant.
The tangent to $C$ at a general point with parameter $t$ cuts the coordinate axes at the points $A$ and $B$.
(i) Denoting the origin by $O$, show that $O A^{\frac{2}{3}}+O B^{\frac{2}{3}}=a^{\frac{2}{3}}$.

[5]

(ii) As $t$ varies, the midpoint of the line segment $A B$ traces out a curve. Find the cartesian equation of this curve in its simplest form.

[3]

(iii) Find, in terms of $a$, the area enclosed by the curve $C$, giving your answer in the form $k a^2$ where $k$ is to be determined correct to 2 decimal places.

[3]