$x=a \sin (\omega t)$ and $y=a(1-\cos (\omega t))$ where $a$ and $\omega$ are constants.

$\Rightarrow \sin ^2(\omega t)=\frac{x^2}{a^2}$

$\Rightarrow \cos ^2(\omega t)=\left(1-\frac{y}{a}\right)^2$

$\Rightarrow \frac{x^2}{a^2}+\left(1-\frac{y}{a}\right)^2=1$

$\Rightarrow x^2+(y-a)^2=a^2$

The given equation is the equation of circle with centre $(0, a)$, the radius a also the angular velocity is $\omega$ so distance covered $=a c o t$