The circles whose equations are $x^2+y^2+c^2=2ax$ and $x^2+y^2+c^2-2by=0$ will touch one another externally, if

The two circles are

$x^2+y^2-2ax+c^2=0$ and $x^2+y^2-2by+c^2=0$

Centres and radii of these two circles are

Centres $\begin{array}{cc}{C}_1(a,0)& C_2(0,b)\\ r_1=\sqrt{a^2-c^2}& r_2=\sqrt{b^2-c^2}\end{array}$

Radii: 

Given that the two circles touch each other externally. 

$\begin{array}{l}\therefore C_1{C}_2=r_1+r_2\\ \Rightarrow \sqrt{a^2+b^2}=\sqrt{a^2-c^2}+\sqrt{b^2-c^2}\\ \Rightarrow a^2+b^2=a^2-c^2+b^2-c^2+2\sqrt{a^2-c^2}\sqrt{b^2-c^2}\\ \Rightarrow c^4=a^2{b}^2-c^2\left(a^2+b^2\right)+c^4\\ \Rightarrow a^2{b}^2=c^2\left(a^2+b^2\right)\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}\end{array}$