How can prove that?
Let C be a collection of disjoint circles in the plane. Show that there is a line that passes through the origin and not any of these tangent circles.
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?