Answer
a) $x=a \cos t; y= -a\sin t$; $0\leq t\leq 2 \pi$
b) $x=a \cos t; y= a\sin t$; $0\leq t\leq 2 \pi$
c) $x=a \cos t; y= -a\sin t$; $0\leq t\leq 4 \pi$
d) $x=a \cos t; y= a\sin t$; $0\leq t\leq 4 \pi$
Work Step by Step
Consider the equation of a circle $x^2+y^2=a^2$ with points $(a,0)$ which leads to the parametric equations: $x=a \cos t; y= a\sin t$
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 $y= a\sin t$ to $y=- a\sin t$. This can be justified with the explanation in parts (a) and (c).
2. When we need to trace the circle twice, 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).
3. When we need to trace the circle counterclockwise to complete one around of a circle, we need $2\pi$ radians so that $0\leq t\leq 2 \pi$. This can be justified with the explanation in the part (b).
Hence, our answers are:
a) $x=a \cos t; y= -a\sin t$; $0\leq t\leq 2 \pi$
b) $x=a \cos t; y= a\sin t$; $0\leq t\leq 2 \pi$
c) $x=a \cos t; y= -a\sin t$; $0\leq t\leq 4 \pi$
d) $x=a \cos t; y= a\sin t$; $0\leq t\leq 4 \pi$
