Do you just have to show existence of such circles?
Here is a problem I've been trying to solve for some time now. Maybe you could help me.
We have two sets
is a set of those circles in the plane such that for any there exists a circle which intersects axis in .
is a set of those circles in the plane such that for any there exists a circle which is tangent to axis in .
That's right. I think that maybe we could somehow identify each circle with a different rational number and this would mean that there is at most countably many disjoint circles in the plane but uncountably many points on x axis, so at least two must intersect. If this reasoning is right, it holds for both sets, doesn't it?