# At least two circles tangent to y axis with nonempty intersection

• Jan 11th 2013, 02:31 AM
wilhelm
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 $\displaystyle (x,0)$.
• Jan 11th 2013, 04:03 PM
chiro
Re: At least two circles tangent to y axis with nonempty intersection
Hey wilhelm.

Do you just have to show existence of such circles?
• Jan 11th 2013, 10:33 PM
wilhelm
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?
• Jan 13th 2013, 09:54 PM
LoidaWard
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?

__________________
Your DVD Zone, Take Your Choice without Any Doubt on The Newsroom Season 1 DVD