The maximum number of points with rational coordinates on a circle whose centre is $$\left( {\sqrt 3 ,\,0} \right)$$  is :

Solution :
There cannot be 3 points on the circle with rational coordinates for then the centre of the circle, being the circumcentre of a triangle whose vertices have rational coordinates, must have rational coordinates ( $$\because $$  the coordinates will be obtained by solving two linear equations in $$x,\,y$$  having rational coefficients ). But the point $$\left( {\sqrt 3 ,\,0} \right)$$  does not have rational coordinates. Also the equation of the circle is $${\left( {x - \sqrt 3 } \right)^2} + {y^2} = {r^2}\,\, \Rightarrow x = \sqrt 3 \pm \sqrt {{r^2} - {y^2}} .$$
For suitable $$r,\,x,$$  where $$x$$ is rational, $$y$$ may have two rational values.
For example, $$r=2,\,x=0,\, y=1,\,-1$$      satisfy $$x = \sqrt 3 \pm \sqrt {{r^2} - {y^2}} .$$
So we get two points $$\left( {0,\,1} \right),\,\left( {0,\, - 1} \right)$$    which have rational coordinates.