Question

How do you determine the parametric equations of the path of a particle that travels the circle: `(x−3)^2+(y−1)^2=9` on a time interval of `0<=t<=2`?

Answer 1

`x =3+ 3 cos pi t`
`y =1 + 3 sin pi t`

Explanation:

`(x−3)^2+(y−1)^2=9`

which is a circle that we can simplify as follows:
`u^2+v^2=3^3`

where `u = x-3, v = y -1`

so

`u = 3 cos psi, v = 3 sin psi implies u^2 + v^2 = 3^3`

`x-3 = 3 cos psi, y-1 = 3 sin psi`

`implies x =3+ 3 cos psi, y =1 + 3 sin psi`

in terms of periodicity, we can say that `psi = omega t` where `omega = (2 pi) /T`

if the period T is `t = 2`, then `omega = (2 pi) /2 = pi`

So
`x =3+ 3 cos pi t`
`y =1 + 3 sin pi t`