A point $P(x, y)$ moves in $XY$ plane such that $x = a\cos^2 \theta$ and $y = 2a \sin \theta$, where $\theta$ is a parameter. The locus of the point $P$ is

$y^2=4a^{2}\sin^{2}{\theta}$

$=4a^{2}-4ax=-4a(x-a)$

Sthe ince $0\le\cos^{2}{\theta}\le{1}$

$\Rightarrow 0\le x \le a$

Also, $-1\le\sin{\theta}\le{1}$

$\Rightarrow -2a\le y \le 2a$

So, the locus of $P$ is a part of parabola.