What is the number of rational points $(a,b)$ (a point is considered to be rational if both $a$ and $b$ are rational) on the circumference of a circle with center as $(\pi,e)$?
My line of reasoning is as follows:-
The equation of the circle can be written as:
$$(x-\pi)^2+(y-e)^2=r^2$$
where $r = \sqrt{(a-\pi)^2+(b-e)^2}$
Now clearly $(a-\pi)$ and $(b-e)$ are irrational individually as $a,b$ are both rational. But can the sum of their squares be possibly rational? At first I thought maybe not but then taking an example,
$$\sqrt{(\sqrt2)^2+(\sqrt2)^2}=2$$
which is kind of the same form $(\sqrt{(Irr)^2+(Irr)^2})$ but the value turns out to be rational.
So, how would I determine the number of rational points on the the circumference? How do I know if the radius of this circle turns out to be rational or irrational??
My book says the answer is "atmost one". I can't really visualize why.