Let n be the number of points having rational coordinates at a fixed distance from the point $(0,\sqrt{3})$. Then

Let $\mathrm{A}({\mathrm{x}}_1,{\mathrm{y}}_1)\mathrm{and}\mathrm{B}({\mathrm{x}}_2,{\mathrm{y}}_2)$ be two points at equal distance a from $\mathrm{P}(0,\sqrt{3}).$Then

${\mathrm{x}}_1^2+{({\mathrm{y}}_1-\sqrt{3})}^2={\mathrm{a}}^2$

and ${\mathrm{x}}_2^2+{({\mathrm{y}}_2-\sqrt{3})}^2={\mathrm{a}}^2$

Subtracting (2) from (1), we get 

${\mathrm{x}}_1^2-{\mathrm{x}}_2^2+{\mathrm{y}}_1^2-{\mathrm{y}}_2^2-2\sqrt{3}({\mathrm{y}}_1-{\mathrm{y}}_2)=0$

Equating irrational and rational parts to zero, we get

${\mathrm{y}}_1={\mathrm{y}}_2$

and ${\mathrm{x}}_1^2-{\mathrm{x}}_2^2+{\mathrm{y}}_1^2-{\mathrm{y}}_2^2=0$

Thus, ${\mathrm{y}}_1={\mathrm{y}}_2=\mathrm{a}\left(\mathrm{say}\right)$

and ${\mathrm{x}}_1^2-{\mathrm{x}}_2^2={\mathrm{b}}^2\left(\mathrm{say}\right)$

Hence, at most two such points are possible whose coordinates are (b,a) and (-b,a). In a special case, when b=0, only one such point is possible.