1. ## please explain how to find the parametric eq.

find parametric equations for the path of particle that moves along x2+(y-1)2=4 in a manner described.....

A.) once around clockwise, starting at (2,1)
B.) three times around counterclockwise, starting at (2,1)
C.) halfway around counterclockwise, starting at (3,0)

2. ## Re: please explain how to find the parametric eq.

Originally Posted by pnfuller
find parametric equations for the path of particle that moves along x2+(y-1)2=4 in a manner described.....

A.) once around clockwise, starting at (2,1)
B.) three times around counterclockwise, starting at (2,1)
C.) halfway around counterclockwise, starting at (3,0)

Do not double post.

3. ## Re: please explain how to find the parametric eq.

Originally Posted by pnfuller
find parametric equations for the path of particle that moves along x2+(y-1)2=4 in a manner described.....

A.) once around clockwise, starting at (2,1)
B.) three times around counterclockwise, starting at (2,1)
C.) halfway around counterclockwise, starting at (3,0)

Equation of this circle with radius=2 and center=(1,0) in complex plane is:

$\displaystyle z(\theta \)\text{=}1+2 e^{i \theta }$

Let t1, t2, t3 be 3 parameters representing stages 1,2,3 of rotation. Then the path is:

$\displaystyle path=\left[e^{-2 \pi i t_1} e^{\left(6 \pi -\frac{\pi }{4}\right) i t_2} e^{\pi i t_3} z\left(\frac{\pi }{4}\right)\right]$

Simplifying we have:

$\displaystyle path= \sqrt{2} \sin \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)+\sqrt{2} \cos \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)+\cos \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)+i \left(-\sqrt{2} \sin \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)-\sin \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)+\sqrt{2} \cos \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)\right)$

Notice that horizontal component of particle's location is real part of path and its verttical component as imaginary part of the path.

$\displaystyle x(t)=\sqrt{2} \sin \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)+\sqrt{2} \cos \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)+\cos \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)$

$\displaystyle y(t)=-\sqrt{2} \sin \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)-\sin \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)+\sqrt{2} \cos \left(2 \pi t_1-\pi t_3-\frac{23 \pi t_2}{4}\right)$