Let both of the ends of the rod are on $x$-axis and $y$-axis. Let $A B$ be rod of length $l$ and coordinates of $A$ and $B$ be $(a, 0)$ and $(0, b)$ respectively.
Let $P(h, k)$ be the mid point of the rod $A B$.
Now, in $\Delta O A B$,
$O A^{2}+O B^{2} =A B^{2}$
$a^{2}+b^{2} =l^{2}$
$\Rightarrow (2 h)^{2}+(2 k)^{2}=l^{2}$ [using Eq. (i)]
$\Rightarrow h^{2}+k^{2}=\frac{l^{2}}{4}$
$\therefore $ The equation of locus is
$x^{2}+y^{2}=\frac{l^{2}}{4}$
