The equation to the locus of the midpoints of chords of the circle $x^2+y^2=r^2$ having a constant length $2l$ is

Let $P(x_1,y_1)$ be the midpoint of the chord and $AB=2l\Rightarrow PB=l$ 

here $\angle OPB=90°,O{B}^2=O{P}^2+P{B}^2$

$\Rightarrow r^2=x_1^2+y_1^2+l^2\Rightarrow x_1^2+y_1^2=r^2-l^2$

Locus of $\text{'}P\text{'}$ is $x^2+y^2=r^2-l^2$