# Circle passing through intersection points of two bigger circles

Suppose the equations of two intersecting circles are given.Now how to find the equation of circle passing through the points of intersection of the larger circles? Now please dont tell me that i got to solve the two equations for point of intersections :-P!! I guess there must be a shorter method..any ideas? P.S. i MEANT THE SMALLEST POSSIBLE CIRCLE.

• Do you mean the smallest one in the pencil or the whole $\mu_1C_1+\mu_2C_2=0$? – Jan-Magnus Økland Mar 29 '15 at 7:39
• Yes the smallest one :-P..im sorry..i forgot to mention. – user220382 Mar 29 '15 at 8:32

Let the equations of two intersecting circles be $C_1=0$ and $C_2=0$.
Then the equation of family of circles passing through the intersection points can be given by $C_1 + tC_2 = 0$, $t\ne -1$. It is easy to see that this equation satisfies the points that are common to both the circles.