Answer
a) $x=a \cos t; y= -b\sin t$; $0\leq t\leq 2 \pi$
b) $x=a \cos t; y= b\sin t$; $0\leq t\leq 2 \pi$
c) $x=a \cos t; y= -b\sin t$; $0\leq t\leq 4 \pi$
d) $x=a \cos t; y= b\sin t$; $0\leq t\leq 4 \pi$
(Other answers are possible.)
Work Step by Step
Consider the equation of an ellipse: $(\dfrac{x}{a})^2+(\dfrac{y}{b})^2=1$ with point $(a,0)$ and parametric equations: $x=a \cos t; y= b\sin t$
Depending upon these parametric equations, we have some common points to be noticed.
1. When we need to trace the circle clockwise, we have to change $y= b \sin t$ to $y=-b\sin t$. This can be justified with the explanation in parts (a) and (c).
2. When we need to trace the circle two times, we have to double $2\pi$ radians to $4 \pi$ radians so that $0\leq t\leq 4 \pi$. This can be justified with the explanation in parts (c) and (d).
Hence, our answers are:
a) $x=a \cos t; y= -b\sin t$; $0\leq t\leq 2 \pi$
b) $x=a \cos t; y= b\sin t$; $0\leq t\leq 2 \pi$
c) $x=a \cos t; y= -b\sin t$; $0\leq t\leq 4 \pi$
d) $x=a \cos t; y= b\sin t$; $0\leq t\leq 4 \pi$
