Answer
$\left\{\begin{array}{l}
x=a\sec\theta\\
y=b\sin\theta
\end{array}\right.$
Work Step by Step
Point P has coordinates $(x_{P}, y_{P}).$
A lies on the outer circle with radius a:
$(x_{A},y_{A})=(a\cos\theta, a\sin\theta)$
The y-coordinate of p is the same as the y-coordinate on the smaller circle,
$ y_{P}=b\sin\theta$
The x-coordinate of P is the x-coordinate of B.
With the right angle at A, $\triangle \mathrm{O}\mathrm{A}\mathrm{B}$ is a right triangle, OB is the hypotenuse,
and $\displaystyle \frac{|OB|}{|OA|}=\sec\theta$, so
$x_{B}=a\sec\theta=x_{P}$
We have
$P=(a\sec\theta,b\sin\theta)$, and the parametric equations are
$\left\{\begin{array}{l}
x=a\sec\theta\\
y=b\sin\theta
\end{array}\right.$
