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

Let A(x1,y1) and B(x2,y2) be two points at equal distance a from P(0,3).Thenx12+(y1−3)2=a2and x22+(y2−3)2=a2Subtracting (2) from (1), we get x12−x22+y12−y22−23(y1−y2)=0Equating irrational and rational parts to zero, we gety1=y2and x12−x22+y12−y22=0Thus, y1=y2=a(say)and x12−x22=b2(say)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.