A curve $C$ has parametric equations $x=a{{\sin }^{5}}t$, $y=a{{\cos }^{5}}t$ where $a$ is a positiv

Question

A curve $C$ has parametric equations $x=a{{\sin }^{5}}t$, $y=a{{\cos }^{5}}t$ 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 $B$. Denoting the origin by $O$, show that $O{{A}^{\frac{2}{3}}}+O{{B}^{\frac{2}{3}}}={{a}^{\frac{2}{3}}}$. [5] As $t$ varies, the midpoint of the line segment $AB$ traces out a curve. Find the cartesian equation of this curve in its simplest form. [3] 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]

Accepted Answer

$OA^{2/3} + OB^{2/3} = a^{2/3}$; Area $= 0.37a^2$

2021 NYJC P1 Q8

2021 NYJC P1 Q8

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