Consider a circle of radius $a = 2$ centered at $(0,a)$, as in the figure. Let a line from the origin $O$ to a point $A$ on the circle intersect the line $y = 2a$ at $B$. Finally, let $C$ be the point of intersection of a horizontal line through $A$ and a vertical line through $B$. As $t$, the angle $OA$ makes with the positive $x$-axis varies, point $C$ traces out a curve called the witch of Agnesi.

(a) Find a vector-parametric equation for the point $A$ in terms of the parameter $t$. Your answer should be of the form $\langle x(t), y(t) \rangle$ and include the angle brackets. $\vec{r}_A(t) =$ (b) Find a vector-parametric equation for the point $B$ in terms of the parameter $t$. $\vec{r}_B(t) =$ (c) Find a vector-parametric equation for the point $C$ in terms of the parameter $t$. $\vec{r}_C(t) =$