91Ó°ÊÓ

The curve starting at the point $(0,3)$with $t∈[0,1]$.

Consider a curve starting at the point $(0,3)$with $t∈[0,1]$.

The objective is to find the parametric equations which represent the given condition.

Given that the curve is centered at the origin ,traced once in clockwise direction.

The curve is a unit circle starting from the point $(0,3)$so the radius of the curve is 3 .

The parametric equations which moves counterclockwise direction starting at $(0,3)$is given by

$(x,y)=(r\sin 2\mathrm{Ï€}t,r\cos 2\mathrm{Ï€}t)$

In the counterclockwise direction, replace t by $-t$.then,

$$\begin{gathered} (x,y)=\left(r\sin 2\mathrm{Ï€}\right(-t),r\cos 2\mathrm{Ï€}(-t\left)\right) \\ (x,y)=(-r\sin 2\mathrm{Ï€}t,r\cos 2\mathrm{Ï€}t) \end{gathered}$$

Here the radius $r=3$then,

$x=-3\sin 2\mathrm{Ï€}t$and $y=3\cos 2\mathrm{Ï€}t$ forms a circle with center $(0,0)$ and radius is 3 which starts at $(0,3)$ moving in counterclockwise direction.

Therefore, the required parametric equations are $x=-3\sin 2\mathrm{Ï€}t,y=3\cos 2\mathrm{Ï€}t$