Question

Given two circles C1 and C2 in the same plane where C1 has a radius of a and C2 has a radius of b and the centers of the C1 and C2 have a distance of c apart. In terms of a,b, and c describe when the intersection of C1 and C2 would be zero points. prove this

Answer 1

In the diagram above, all of the three possible cases have been drawn. In the first case, when c < a + b, then there are infinite intersection points. In the second case, when c = a + b, then we have one intersection point. And lastly, when c > a + b, has no intersection, i.e the intersection has zero points. This can be proved as follows:

We just need to consider third case : Suppose circle with center X has radius b and circle with center Y has radius a. Assume p lies in both the circles. So, d(X,p) b; d(Y,p) a. By triangle inequality, we have d(X,Y) d(X,p) + d(Y,p). And so we get d(X,Y) a + b. As the distance between the circles C1 and C2 is c, c a + b. Hence if

c > a + b, then there is no such p, that is, the intersection of C1 and C2 would be zero points. where d is distance between points.