P is a point $(a, b)$ in the first quadrant. If the two circles which pass through P and touch both the coordinate axes cut at right angles, then

P(a, b) and eq of 1st circle is
${\mathrm{x}}^2+{\mathrm{y}}^2-2{\mathrm{r}}_1\mathrm{x}-2{\mathrm{r}}_1\mathrm{y}+{\mathrm{r}}_1^2=0\to \left(1\right)$
Eq of (2^nd) circle is
${\mathrm{x}}^2+{\mathrm{y}}^2-2{\mathrm{r}}_2\mathrm{x}-2{\mathrm{r}}_2\mathrm{y}+{\mathrm{r}}_2^2=0\to \left(2\right)$

(1) & (2) interest orthogonally, $4{\mathrm{r}}_1{\mathrm{r}}_2={\mathrm{r}}_1^2+{\mathrm{r}}_2^2$
and common chord (1) & (2) passes through (a, b) is
$\begin{array}{l}2(\mathrm{a}+\mathrm{b})={\mathrm{r}}_1+{\mathrm{r}}_2\text{ and }{\mathrm{r}}_1{\mathrm{r}}_2={\mathrm{a}}^2+{\mathrm{b}}^2\\ \therefore {\mathrm{a}}^2-4\mathrm{ab}+{\mathrm{b}}^2=0\end{array}$