# 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

## 1 Answer

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.

• Hey your answer is perfect..but i needed the smallest circles's equation. – user220382 Mar 29 '15 at 8:33
• Ohk.. now that looks interesting, let me think a bit.. – AgentS Mar 29 '15 at 8:39
• Easy : the largest chord of a circle is diameter and the minimum length for diameter is achived when the ends are intersecting points of the given circles. So you want diameter to be the segment connecting the intersecting points. – AgentS Mar 29 '15 at 8:43
• Right..but what will be the method to find diameter?Any idea? – user220382 Mar 29 '15 at 8:55
• I really dont want to calculate the intersecting points..that will be a very ugly method..i feel a much more elegant solution exists..and im still trying to figure it out – user220382 Mar 29 '15 at 8:56