Let $P\left(h,k\right)$ be the point dividing te line segment in the ratio $2:3$, then

$h=\frac{2\left(10\cos \theta \right)+3\left(5\right)}{2+3}=4\cos \theta +3$ and $k=\frac{2\left(10\sin \theta \right)+3\left(0\right)}{2+3}=4\sin \theta$
$\therefore {\left(h-3\right)}^2+k^2=16$
$\Rightarrow$ Locus of $P\left(h,k\right)$ is ${\left(x-3\right)}^2+y^2=16$, which is a circle.

Hence, the perimeter of the circle is $2\mathrm{\pi }\left(4\right)=8\mathrm{\pi }$ units