The lengths of the intercepts made by any circle on the coordinates axes are equal if the centre lies on the line $\left(s\right)$ represented by

Let the equation of any circle be
$x^2+y^2+2gx+2fy+c=0$                (i)

For intercept made by the circle on $x-$axis, put $y= 0$ in (i)

$\Rightarrow x^2+2gx+c=0$

If $x_1, x_2$ are roots of (ii), then length of the intercept on $x-$axis is

$\left|x_1-x_2\right|=\sqrt{{\left(x_1+x_2\right)}^2-4x_1{x}_2}=2\sqrt{g^2-c}$

Similarly length of the intercept of the $y-$axis is $2\sqrt{f^2-c}$

Since the lengths of these intercepts are equal

$\begin{array}{l}\sqrt{g^2-c}=\sqrt{f^2-c}\\ \Rightarrow g^2=f^2=(-g{)}^2=(-f{)}^2\end{array}$

Therefore, centre lies on $x^2 – y^2 = 0$