A stick of length $l$  moves on coordinate axes. The Locus of midpoint of the stick is

$Let\frac{x}{a}+\frac{y}{b}=1$  meets $x-axis$ at $A\left(a,0\right)$ 

$y-axis$  at $B\left(o,b\right)$

$AB=l,$  $\Rightarrow a^2+b^2=l^2$ ___________(1)

Locus point $\left(x,y\right)=midpo\mathrm{int}ofAB$

$\left(x,y\right)=\left(\frac{a}{2},\frac{b}{2}\right)$

$\Rightarrow a=2x,b=2y$

Substitute  in (1)  $\Rightarrow 4x^2+4y^2=l^2$