If a point P is moving such that the lengths of tangents drawn from P to the circles ${\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}-6\mathrm{y}-12=0$ and ${\mathrm{x}}^2+{\mathrm{y}}^2+6\mathrm{x}+18\mathrm{y}+26=0$ are in the ratio 2:3, then find the equation to the locus of P.

Given circles are
$\mathrm{S}={\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}-6\mathrm{y}-12=0 ...\left(1\right)$
${\mathrm{S}}^{\mathrm{\prime }}={\mathrm{x}}^2+{\mathrm{y}}^2+6\mathrm{x}+18\mathrm{y}+26=0 ...\left(2\right)$
Let P(x₁, y₁) be a point on the locus
$\text{ Given }\frac{\sqrt{{\mathrm{S}}_{11}}}{\sqrt{{\mathrm{S}}_{11}^1}}=\frac{2}{3}\Rightarrow 9{\mathrm{S}}_{11}=4{\mathrm{S}}_{11}^1$
$\begin{array}{l}\Rightarrow 9\left({\mathrm{x}}_1^2+{\mathrm{y}}_1^2-4{\mathrm{x}}_1-6{\mathrm{y}}_1-12\right)=4\left({\mathrm{x}}_1^2+{\mathrm{y}}_1^2+6{\mathrm{x}}_1+18{\mathrm{y}}_1+26\right)\\ \Rightarrow 5\left({\mathrm{x}}_1^2+{\mathrm{y}}_1^2\right)-60{\mathrm{x}}_1-126{\mathrm{y}}_1-212=0\end{array}$
$\text{ Locus of }\mathrm{P}\text{ is }5\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)-60\mathrm{x}-126\mathrm{y}-212=0$