If two circles which pass through the points  (a, 0) and (– a, 0) touch the line y = mx + c and cut
orthogonally then

Centres of the circles lie on y-axis as x-axis is the common chord. Let the centres be $\left(0,f_2\right)$$\left(0,f_1\right)$ and circles intersect orthogonally, so $\frac{f_1}{a}\times \frac{f_2}{a}=-1$

$\Rightarrow f_1{f}_2=-a^2$.

As they touch $y=mx+c$.

$\frac{f_1-c}{\sqrt{1+m^2}}=a^2+f_1^2$

$\Rightarrow m^2{f}_1^2+2c{f}_1+\left(1+m^2\right)a^2-c^2=0$ which gives two values of $f_1$ say $f_1$and$f_2$such that

$\begin{array}{l}{f}_1{f}_2=\frac{\left(1+m^2\right)a^2-c^2}{m^2}=-a^2\\ \Rightarrow c^2=\left(2m^2+1\right)a^2\end{array}$