Math Help - 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!

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 are any two points the locus of points that are centers of circles through is the perpendicular bisector of . Any line has uncountably many points.
For you question about , there is no reason to exclude the point.