The combined equation of the pair of tangents drawn from an external point $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\text{ to the circle }\mathrm{S}=0\text{ is }{\mathrm{S}}_1^2={\mathrm{SS}}_{11}$

Suppose that the tangents drawn from P to the circle S = 0 touch the circle at A and B.
The equation of AB is ${\mathrm{S}}_1=0$
$\text{ i.e., }{\mathrm{xx}}_1+{\mathrm{yy}}_1+\mathrm{g}\left(\mathrm{x}+{\mathrm{x}}_1\right)+\mathrm{f}\left(\mathrm{y}+{\mathrm{y}}_1\right)+\mathrm{c}=0\text{. ...(1)}$

$\text{ Let }\mathrm{Q}\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)\text{ be any point on these tangents. }$
The locus of Q will be the equation of pair of tangents drawn from P.
$\text{ The segment }\overline{\mathrm{PQ}}\text{ is divided by the line }\mathrm{AB} \text{ whose equation is }{\mathrm{S}}_1=0\text{ in the ratio }-{\mathrm{S}}_{11}:{\mathrm{S}}_{12}$
$\begin{array}{l}\therefore \mathrm{PB}:\mathrm{QB}=-{\mathrm{S}}_{11}:{\mathrm{S}}_{12}\text{ or }\\ {\mathrm{S}}_{11}:{\mathrm{S}}_{12}\text{ according as }\\ {\mathrm{S}}_{11}:{\mathrm{S}}_{12}<0\text{ or }{\mathrm{S}}_{11}:{\mathrm{S}}_{12}>0\\ \Rightarrow \frac{\mathrm{PB}}{\mathrm{QB}}=\left|\frac{{\mathrm{S}}_{11}}{{\mathrm{S}}_{12}}\right| ...\left(2\right)\end{array}$
$\text{ But }\mathrm{PB}=\sqrt{{\mathrm{S}}_{11}}\text{ and }\mathrm{QB}=\sqrt{{\mathrm{S}}_{22}}\text{ (lengths of the tangents from P and Q)}$
$\text{ (2) }\Rightarrow \frac{{\mathrm{S}}_{11}^2}{{\mathrm{S}}_{12}^2}=\frac{{\mathrm{S}}_{11}}{{\mathrm{S}}_{22}}\Rightarrow {\mathrm{S}}_{11}\cdot {\mathrm{S}}_{22}={\mathrm{S}}_{12}^2$
$\text{ Locus of }\mathrm{Q}\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)\text{ is }{\mathrm{SS}}_{11}={\mathrm{S}}_1^2\text{. }$