Let the equation of circle $S=0$, passes through
the point $(0,0)$ is
$S=x^2+y^2+2 g x+2 f y=0$
Since, circle $S=0$ touches the line $y=x$, so
$\begin{aligned} & \frac{|-g+f|}{\sqrt{2}}=\sqrt{g^2+f^2} \\ & \Rightarrow \quad g^2+f^2-2 g f=2\left(g^2+f^2\right) \\ & \Rightarrow \quad g^2+f^2+2 g f=0 \\ & \Rightarrow \quad g+f=0 \\ & \end{aligned}$
So, the equation of circle $S=0$, becomes
$x^2+y^2+2 g x-2 g y=0$
Now, the equation of common chord of circles (ii) and $x^2+y^2+6 x+8 y-7=0$ is
$\begin{aligned} & (2 g-6) x-(2 g+8) y+7=0 \\ \Rightarrow \quad & 2 g(x-y)-(6 x+8 y-7)=0\end{aligned}$
Eq. (iii) represents the family of lines, passes through the intersection of lines
$x-y=0$ and $6 x+8 y-7=0$
And the point of intersection is $\left(\frac{1}{2}, \frac{1}{2}\right)$. Hence, option (a) is correct.