If $S=0$ bisects the circumference of $S^{\prime}=0$ then the common chord will pas thoo ufh the $ \text { centre of } S^{\prime}=0 \text { . } $

$S=0$

$x^{2}+y^{2}+2 g x+2 f y+c=0$

$x^{2}+y^{2}-1=0$

$c \equiv(0,0)$

$S=0 \quad$ and $\quad S_{1}=0$

Common chord $\Rightarrow \quad S-S_{1}=0$

$\left(x^{2}+y^{2}+2 g x+2 f y+c\right)-\left(x^{2}+y^{2}-1\right)=0$

$2 g x+2 f y+c+1=0$

$\Rightarrow C+1=0$

$\Rightarrow C=-1$

$S=0$

$x^{2}+y^{2}+2 g x+2 f y-1=0 \quad, \quad(-9,-f)$

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

$c^{\prime} \equiv(-1,0)$

$Eq ^{n} $ common $\quad$ chord $\quad \Rightarrow \quad\left(z^{2}+y^{k}+2 g x+2 f y-1\right)-\left(z^{2}+y^{2}+2 x-3\right)=0$

$\Rightarrow \quad 2 g x+2 f y-1-2 x+3=0$

$\Rightarrow \quad-2 g+0-1-2(-1)+3=0$

$\overrightarrow{7} \quad-2 g-1+2+3=0$

$\Rightarrow \quad-2 g=-4$

$\overrightarrow{7} \quad g=2$

$S=0$

$x^{2}+y^{2}+4 x+2 f y+(-1)=0$

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

$c^{\prime \prime} \equiv(0,-1)$

$Eq ^{n}  common \quad$ chord $\quad \Rightarrow \quad\left(x^{2}+y^{2}+4 x+2 f y-1\right)-\left(x^{2}+y^{2}+2 y-3\right)=0$

$\Rightarrow \quad 4 x+2 f y-1-2 y+3=0$

$\Rightarrow \quad 0 \quad-2 f-1+2+3=0$

$\Rightarrow \quad-2 f=-4$

$\Rightarrow \quad i \overline{f=2}\rangle$

(-2,-2) is the centre of the requised circle