If A and B are conjugate point w.r.t a circle with centre ‘O’ and radius ‘r’, then $A{B}^2$ =

$\mathrm{A} \left({\mathrm{x}}_1, {\mathrm{y}}_1\right) \mathrm{B} \left({\mathrm{x}}_2, {\mathrm{y}}_2\right)$ are conjugate points then

 $$\begin{gathered} S_{12}=0 \\ \Rightarrow x_1{x}_2+y_1{y}_2={\mathrm{r}}^2 \\ Now {\left(\mathrm{OA}\right)}^2+{\left(\mathrm{OB}\right)}^2-2{\mathrm{r}}^2= \\ ={x^2}_1+{y_1}^2+{x_2}^2+{y_2}^2-2(x_1{x}_2+y_1{y}_2) \\ ={(x_1-x_2)}^2+{(y_1-y_2)}^2 \\ ={\left(\mathrm{AB}\right)}^2 \end{gathered}$$