The length of the common chord of the circles $x^2+y^2+4x+1=0$and $x^2+y^2+4y-1=0,$ is

Given circles intersect orthogonally. So, the length of their common chord is

        $l=\frac{2r_1{r}_2}{\sqrt{r_1^2+r_2^2}},$

where $r_1$ and $r_2$ are the radii of the given circles.

Here, $r_1=\sqrt{5}$ and $r_2=\sqrt{3}\text{. }$

$\therefore l=\frac{2\sqrt{15}}{\sqrt{5+3}}=\sqrt{\frac{15}{2}}..$