At least two circles tangent to y axis with nonempty intersection

Hi.

Here is a problem I've been trying to solve for some time now. Maybe you could help me.

We have two sets

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

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

Re: At least two circles tangent to y axis with nonempty intersection

Hey wilhelm.

Do you just have to show existence of such circles?

Re: At least two circles tangent to y axis with nonempty intersection

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?

Re: At least two circles tangent to y axis with nonempty intersection

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

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