Answer
$\left\{\begin{array}{l}
x=2a\cot\theta\\
y=2a\sin^{2}\theta
\end{array}\right.$
Work Step by Step
Point P has coordinates $(x_{P}, y_{P})=(x_{C},y_{A}).$
A lies on a circle centered at (0,a), with radius a.
Name the point (0,2a) as B, the endpoint of the diameter through the origin.
Name the point ($x_{A},0)$ as D, on the x axis.
C has coordinates $(x_{C},y_{C})$
From the right triangle $\triangle OBC$, where $\angle \mathrm{O}\mathrm{C}\mathrm{B}=\theta$ (alternate angles),
$\displaystyle \frac{x_{C}}{\text{diameter}}=\cot\theta \Rightarrow x_{C}=2a\cot\theta\quad=x_{P}$
The angle $\angle OAB$ is a right angle, (Thales' theorem)
and the triangle $\triangle OAB$ a right triangle. We have
$\angle AOB=90^{o}-\theta, \quad$ and $\angle OBA=\theta.$
$\displaystyle \sin\theta=\frac{|OA|}{2a}\Rightarrow|OA|=2a\sin\theta$
From the triangle $\triangle ODA,$
$ y_{A}=|OA|\sin\theta\quad$ ... substitute OA...
$y_{A}=2a\sin^{2}\theta=y_{P}.$
So, $P=(2a\cot\theta, 2a\sin^{2}\theta )$ and the parametric equations are
$\left\{\begin{array}{l}
x=2a\cot\theta\\
y=2a\sin^{2}\theta
\end{array}\right.$
To sketch, build a table of function values for x(t) and y(t),
plot the points (x(t), y(t)) as t increases.
