Parametric Equations and Curves — Question 8

Question 8

Problem

A moving point satisfies $x=2+3\cos t$, $y=-1+3\sin t$. Without eliminating $t$, determine its center, radius, starting point, and initial direction.

Question 8 – Solution

Solution

1. Compare the equations $$x=2+3\cos t, \qquad y=-1+3\sin t$$ with the standard circle parametrization $$x=h+R\cos t, \qquad y=k+R\sin t.$$

2. It follows immediately that $$(h,k)=(2,-1) \qquad\text{and}\qquad R=3.$$ Thus the curve is a circle centered at $\boxed{(2,-1)}$ with radius $\boxed{3}$.

3. Evaluate the coordinates at $t=0$: $$x(0)=2+3\cos0=5, \qquad y(0)=-1+3\sin0=-1.$$ Therefore, the starting point is $\boxed{(5,-1)}$.

4. Differentiate to find the velocity: $$\mathbf r'(t)=(-3\sin t,3\cos t).$$ At $t=0$, $$\mathbf r'(0)=(0,3),$$ which points straight upward from the rightmost point of the circle.

5. Upward motion from the rightmost point is counterclockwise, so the initial direction is .