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) $x=a \cos t; y= b\sin t$; $0\leq t\leq 4 \pi$
Work Step by Step
The equation of a circle is $(\dfrac{x}{a})^2+(\dfrac{y}{b})^2=1$ at point $(a,0)$ with parametric equations such as: $x=a \cos t; y= b\sin t$
These parametric equations tells us the following some points:
I. When we need to trace the circle clockwise, the parametric equation changes from $y= b \sin t$ to $y=-b\sin t$. This leads to the explanation given in the parts (a) and (c).
II. When we need to trace the circle twice times, we will have to double the angle $2\pi$ radians to $4 \pi$ radians so that $0\leq t\leq 4 \pi$. This leads to the explanation given in the parts (c) and (d).
III. When we will trace the circle counterclockwise to complete a one around of a circle, we will have to change the angle to$2\pi$ radians so that $0\leq t\leq 2 \pi$. This leads to the explanation given in the part (b).
