Geometry: System of Circles

Question

Given a circle $\,x^2+y^2+dx+ey+c=0,\,$ find the general equation of a circle passing through the intersection of this circle and the line $\,lx+my+n=0.$

My approach was to consider a circle of the same form $\,x^2+y^2+Dx+Ex+C=0\,$ and then use the fact that $\,x^2+y^2+dx+ey+c+K(x^2+y^2+Dx+Ex+C)=0\,$ By setting $\,K=-1,\,$ I would get the equation of the line given above and then I could compare the variables $D$, $E$, $C$ to get the parameterized equation of the circle. Is the approach correct? Is there another way to solve this problem elegantly?

Thank you.

Answer 1

If $S=0$ is the circle and $L=0$ is the line, then the curve $S+tL=0$ for all non-zero values of $t$ will represent a circle passing through the intersection of $S=0$ and $L=0$.

Answer 2

Another approach, HINT

Combining $\,x^2+y^2+dx+ey+c=0,\,$ and $\,lx+my+n=0$ will give the coordinates of the two intersection points (for suitable parameters). The perpedicular bisector of the segment joining the intersection points is the location of the centers of our circles. We need those that go through the two intersection points found above.