Find the pole of ax + by + c = 0w.r.t x² + y²= r².

$\text{ Let }\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\text{ be the pole of }$
$\mathrm{ax}+\mathrm{by}+\mathrm{c}=0\dots \left(1\right)\text{ w.r.t }{\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{r}}^2$
$\text{ Equation of polar of }\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ 
wrt the circle is ${\mathrm{S}}_1=0$
$\Rightarrow {\mathrm{xx}}_1+{\mathrm{yy}}_1-{\mathrm{r}}^2=0$
(1) & (2) represent same line, polar
$\begin{array}{l}\therefore \frac{{\mathrm{x}}_1}{\mathrm{a}}=\frac{{\mathrm{y}}_1}{\mathrm{b}}=\frac{-{\mathrm{r}}^2}{\mathrm{c}}\\ \Rightarrow \frac{{\mathrm{x}}_1}{\mathrm{a}}=\frac{-{\mathrm{r}}^2}{\mathrm{c}}\Rightarrow \frac{{\mathrm{y}}_1}{\mathrm{b}}=\frac{-{\mathrm{r}}^2}{\mathrm{c}}\\ \Rightarrow {\mathrm{x}}_1=\frac{-{\mathrm{r}}^2\mathrm{a}}{\mathrm{c}}\Rightarrow {\mathrm{y}}_1=\frac{-{\mathrm{r}}^2\mathrm{b}}{\mathrm{c}}\\ \therefore \text{ pole }\mathrm{p}=\left(\frac{-{\mathrm{r}}^2\mathrm{a}}{\mathrm{c}},\frac{-{\mathrm{r}}^2\mathrm{b}}{\mathrm{c}}\right)\end{array}$