$x^2+y^2+x+3y=0$

$Ab:x+y+1=0$ is chord of contact from intersection point $\left(h,k\right)$

$AB; xh+yk+\frac{x+h}{2}+3\left(\frac{y+k}{2}\right)=0$

$\Rightarrow x(h+\frac{1}{2})+y(k+\frac{3}{2})+\frac{h}{2}+\frac{3k}{2}=0$

On comparing

$\frac{h+{\displaystyle \frac{1}{2}}}{1}=\frac{k+{\displaystyle \frac{3}{2}}}{1}=\frac{{\displaystyle \frac{h}{2}+\frac{3k}{2}}}{1}$

$h+\frac{1}{2}=k+\frac{3}{2}$

$h-k=1 ...\left(1\right)$

$h+\frac{1}{2}=\frac{h}{2}+3\frac{k}{2}$

$\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\frac{3k}{2}=-\frac{1}{2}$

$h-3k=-1 ...\left(2\right)$

$1, 2$

$2k=2\Rightarrow k=1, h=2$

Circle passing through $A \& B$ is

$x^2+y^2+x+3y+\partial (x+y+1)=0$

It passes through $(2,1)$

$4+1+2+3+\partial \left(4\right)=0$

$\partial =-\frac{5}{2}$

$x^2+y^2-\frac{3x}{2}+\frac{y}{2}-\frac{5}{2}=0$

Centre $(\frac{3}{4}, -\frac{1}{4})$