When will there be a dense subset of rational points in a circle?
If $x^2+y^2=r^2$ has any rational point, then the rational points in it are dense in it.
More generally, the following are equivalent about a (non-degenerate) circle in $\mathbb R^2:$
- The set of rational points on a circle are dense in the circle
- The circle has a rational center and a rational point
- The circle has three rational points.
I'll outline why $2\implies 1.$ We can assume that the center of your circle is $(0,0).$ Your circle has an equation like:
Since it has a rational point, it also means $r^2$ is rational.
Now, if $(x_1,y_1)$ is your rational point, take any line through that point with a rational slope, $m.$ Then the set of pairs $(x_1+t,y_1+mt)$ will (except when $m$ is the tangent of the circle at $(x_1,y_1)$) hit the circle again. But that yields a rational quadratic equation fo $t$ with a known rational root, $t=0.$ So the other root $t$ is also rational, and the other point is rational.
$3\implies 2$ is because finding the circumcenter of three points is a linear process.
And $1\implies 3$ because $(1)$ means there are infinitely many rational points on the circle, so at least $3.$