Answer
$x=a(sin\theta cos\theta+cot\theta)$,
$y=a(1+sin^2\theta)$
Work Step by Step
Use the figure given in the Exercise:
Step 1. Find the length of OQ: knowing the diameter $2a$ and angle $\theta$, we have $OQ=2a\cdot sin\theta$
Step 2. Find the length of OR: since $sin\theta=\frac{2a}{OR}$, we have $OR=\frac{2a}{sin\theta}$
Step 3. Find the length of OP: since $QR=OR-OQ$ and $QP=QR/2$, we have $OP=OQ+QR/2=2a\cdot sin\theta+(\frac{2a}{sin\theta}-2a\cdot sin\theta)/2=a\cdot sin\theta+\frac{a}{sin\theta}$
Step 4. The parametric equations for the curve of P are: $x=OP\cdot cos\theta$ and $y=OP\cdot sin\theta$
Step 5. Combine the results from steps 3 and 4, we have
$x=a\cdot sin\theta cos\theta+a\cdot cot\theta$,
$y=a\cdot sin^2\theta+a$
