Q. 37 - Calculus

Question

The curve is a circle centered at the origin. It is traced once, counterclockwise, starting at the point $(0,3)$ with $t\in [0,1]$.

Step-by-Step Solution

Answer

The parametric equations are$x=-3\sin 2\pi t,y=3\cos 2\pi t$

Step 1: Given information

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

Step 2: Calculation

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

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

$(x,y)=(r\sin 2\pi t,r\cos 2\pi t)$

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

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

Here the radius $r=3$ then,

$(x,y)=(-3\sin 2\pi t,3\cos 2\pi t)$

$x=-3\sin 2\pi t$and $y=3\cos 2\pi t$ 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 $x=-3\sin 2\pi t,y=3\cos 2\pi t$