Question

Find the maximum number of rational points ( a point $(a,b)$ is rational, if a and $b$ both are rational numbers) on the circumference of a circle having centre $(\pi ,e)$.

Answer 1

Answer: 2

Centre of circle $(p,e)$
If there is a point $(x,y)$ on the circle $(x-\pi {)}^2+(y-e{)}^2=r^2$
Clearly, there are two points on this circle.
For instance, let us assume that there are $3$ such points say $(P,Q,R)$. The perpendicular bisectors of $PQ$ and $QR$ would meet at the centre of the circle which must be a rational point.

Hence the maximum number of such points is$2$