1. ## Countability problem

Hi,

I've got a question on countability like this:
Let S be the set of all circles on the coordinate plane that pass through (1,1) and another point (x*sqrt(2),x*sqrt(2)) for some x is a rational number. Determine if S is a countable set or not.

I saw someone do it by fixing x=0, using perpendicular bisector of the line joining (0,0) to (1,1) as the centers of circles. Then, S will contain uncountably many circles passing through (0,0) and (1,1). But he specifically exclude the point (0.5,0.5) as a center of one of those circles, why is that?
Note: My main focus is why the point (0.5,0.5) has to be excluded as a center of those circles?

Sorry that I know it is a bit too long, but I've tried my best to condense it already. It is quite urgent for me, please help me if you know the answer.
Thank you!

2. ## Re: Countability problem

Originally Posted by wesleybrown
I've got a question on countability like this:
Let S be the set of all circles on the coordinate plane that pass through (1,1) and another point (x*sqrt(2),x*sqrt(2)) for some x is a rational number. Determine if S is a countable set or not.
I saw someone do it by fixing x=0, using perpendicular bisector of the line joining (0,0) to (1,1) as the centers of circles. Then, S will contain uncountably many circles passing through (0,0) and (1,1). But he specifically exclude the point (0.5,0.5) as a center of one of those circles, why is that?
Note: My main focus is why the point (0.5,0.5) has to be excluded as a center of those circles?
This is a curious question. If $\displaystyle P~\&~Q$ are any two points the locus of points that are centers of circles through $\displaystyle P~\&~Q$ is the perpendicular bisector of $\displaystyle \overline{PQ}$. Any line has uncountably many points.
For you question about $\displaystyle (0.5,0.5)$, there is no reason to exclude the point.