Find the equation of the circle passing through the points of intersection of circles
${\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}-6\mathrm{y}+21=0,{\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-15=0$ and (1, 2).

Given circles are $\mathrm{S}={\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}-6\mathrm{y}+21=0 \mathrm{and} S^1={\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-15=0$
Required circles passes through the points of intersection of s = 0 and s^| = 0
$\therefore$Equation of required circle is $S+\lambda S^{\mathrm{\prime }}=0$

$\Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}-6\mathrm{y}+21+\mathrm{\lambda }\left({\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-15\right)=0$
It passes through the point (1, 2).
$\begin{array}{l}\therefore 1^2+2^2-8\left(1\right)-6\left(2\right)+21+\mathrm{\lambda }\left(1^2+2^2-2\left(1\right)-15\right)=0\\ \Rightarrow 1+4-8-12+21+\mathrm{\lambda }(1+4-2-15)=0\\ \Rightarrow 6+\mathrm{\lambda }(-12)=0\Rightarrow \mathrm{\lambda }=\frac{1}{2}\end{array}$
Put in (1) Required circle is
$\begin{array}{l}{\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}-6\mathrm{y}+21+\frac{1}{2}\left({\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-15\right)=0\\ \Rightarrow 2{\mathrm{x}}^2+2{\mathrm{y}}^2-16\mathrm{x}-12\mathrm{y}+42+{\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-15=0\\ \Rightarrow 3{\mathrm{x}}^2+3{\mathrm{y}}^2-18\mathrm{x}-12\mathrm{y}+27=0\end{array}$