Answer
$$c(t)=(x(t),y(t))=(1+2\cos t,1+2\sin t).$$
$(x,y)=(0, 1\pm \sqrt 3)$, $(x,y)=( 1\pm \sqrt 3,0)$.
Work Step by Step
The parametric equations for the circle of radius 2 with center $(1, 1)$ are
$$c(t)=(x(t),y(t))=(1+2\cos t,1+2\sin t).$$
To find the points of intersection of the circle with the x-axis, we put $1+2\cos t=0$. Then we get $t=2\pi/3$ and hence $(x,y)=(0, 1\pm \sqrt 3)$.
To find the points of intersection of the circle with the y-axis, we put $1+2\sin t=0$. Then we get $t=7\pi/6$ and hence $(x,y)=( 1\pm \sqrt 3,0)$.
