Answer
See the full explanation below:
Work Step by Step
(a)
From the graph it can be seen that
$\begin{align}
& \sin t=\frac{PB}{a} \\
& =\frac{XA}{a}
\end{align}$
$XA=a\sin t$
And the length OA is equal to the length of arc PA
The length of arc PA is $at$.
Now,
$\begin{align}
& x=OA-xA \\
& =at-a\sin t \\
& =a\left( t-\sin t \right)
\end{align}$
Thus, the parametric equation for x is $x=a\left( t-\sin t \right)$.
(b)
$\begin{align}
& \cos t=\frac{BC}{a} \\
& BC=a\cos t
\end{align}$
Also, $AC=a$
Then,
$\begin{align}
& y=AC-BC \\
& =a-a\cos t \\
& =a\left( 1-\cos t \right)
\end{align}$
Thus, the parametric equation for y is given by $y=a\left( 1-\cos t \right)$.
