Let $l$ be the length of the rod and $\mathrm{\theta }$, the angle of rod with $x$-axis (horizontal) at some instant of time. Coordinates of the centre of rod at this instant of time are



$x =\frac{l}{2}\cos \theta$ and $y=\frac{l}{2}\sin \theta$

Squaring and adding both these equations, we get:

$x^2+y^2=\frac{l^2}{4}$

This is an equation of a circle of radius $\frac{l}{2}$ and centre at origin.