At least two circles tangent to y axis with nonempty intersection

**wilhelm** · January 11th 2013, 02:31 AM

Hi.

Here is a problem I've been trying to solve for some time now. Maybe you could help me.
We have two sets
$\mathcal {Q}$ is a set of those circles in the plane such that for any $x \in \mathbb{R}$ there exists a circle $O \in \mathcal {Q}$ which intersects $x$ axis in $(x,0)$ .

$\mathcal {T}$ is a set of those circles in the plane such that for any $x \in \mathbb{R}$ there exists a circle $O \in \mathcal {T}$ which is tangent to $x$ axis in $(x,0)$ .

**chiro** · January 11th 2013, 04:03 PM

Hey wilhelm.

Do you just have to show existence of such circles?

**wilhelm** · January 11th 2013, 10:33 PM

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?

**LoidaWard** · January 13th 2013, 09:54 PM

If this reasoning is right, it holds for both sets, doesn't it?

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