\[r=2 sin \theta\]
is a circle radius 1 with centre \[(0,1)\]
.To see this let
\[x=rcos \theta = 2sin \theta cos \theta , \; y=r sin \theta = 2 sin^2 \theta \]
.Then
\[\begin{equation} \begin{aligned} x^2+(y-1)^2 &= (2sin \theta cos \theta)^2+(2 sin^2 \theta -1)^2 \\ &= 4sin^2 \theta cos^2 \theta + 4 sin^4 \theta - 4 sin^2 \theta +1 \\ &= 4sin^21 \theta (cos^2 \theta +sin^2 \theta )-4 sin^2 \theta +1 \\ &= 4 sin^2 \theta - 4 sin^2 \theta +1 \\ &=1 \end{aligned} \end{equation}\]