Find the equation of the circle which passes through the point ( 1 , 2 ) and the points of intersection of…

Let $S_1=x^2+y^2-8x-6y+21=0 \& S_2=x^2+y^2-2x-15=0$

Using family of circles $\left(S_1+\lambda S_2=0\right)$

$x^2+y^2-8x-6y+21+\lambda \left(x^2+y^2-2x-15\right)=0$

Since it passes through $\left(1,2\right)$

$1+4-8-6\left(2\right)+21+\lambda \left(1+4-2\left(1\right)-15\right)=0$

$26-20+\lambda \left(5-17\right)=0$

$6+\lambda \left(-12\right)=0$

$\lambda =\frac{1}{2}$

So, the equation is

$x^2+y^2-8x-6y+21+\frac{1}{2}\left(x^2+y^2-2x-15\right)=0$

$\Rightarrow 3x^2+3y^2-18x-12y+27=0$