I think that by disjoint circles the OP means that to circles share neither a point nor an interior point, in the sense of disks.
If that is the case, then the collection is countable. WHY?
If a circle in the plane does not contain the origin as a point or interior point then there is at least one and at most two tangents to the circle that contain the origin.
There are uncountably many lines through the origin.
Can you finish?