The length of the common chord of the circles x 2 + y 2 + 3 x + 5 y + 4 = 0 and x 2 + y 2 + 5 x + 3 y + 4 =…

$S\equiv x^2+y^2+3x+5y+4=0$ and $S\text{'}\equiv x^2+y^2+5x+3y+4=0$

Equation of common chord is $S-S\text{'}=0$

$\Rightarrow -2x+2y=0$

$\Rightarrow x-y=0$

Centre of $S=0$ is $C\left(\frac{-3}{2},\frac{-5}{2}\right)$

Radius, $r=\sqrt{\frac{9}{4}+\frac{25}{4}-4}=\sqrt{\frac{9}{2}}$

$p=$ length of the perpendicular from $C$ on $AB=$$\frac{\left|{\displaystyle \frac{-3}{2}}+{\displaystyle \frac{5}{2}}\right|}{\sqrt{1+1}}=\frac{1}{\sqrt{2}}$

$AB=$length of common chord$=$$2\sqrt{r^2-p^2}=2\sqrt{\frac{9}{2}-\frac{1}{2}}=2\sqrt{4}=4$