Question

The number of rational point(s) [a point $(a,b)$ is called rational, if $a$ and $b$ both are rational numbers] on the circumference of a circle having centre $(\pi ,e)$ is ?

• A. at most one
• B. at least two
• C. exactly two
• D. infinite

Answer 1

The correct option is C exactly two

C. Exactly two

Can you find a circle centered at P with 2 rational points on its circumference? It turns out you can iff π and e are linearly dependent over Q; that is, iff there exist rational numbers p, q, and r such that pπ+qe=r. To see this, first assume such rational numbers exist. Then, note that (π,e) is equidistant from the two points (q,p+rq) and (−q,−p+rq); indeed, the perpendicular bisector of these two points is simply the line px+qy=r, which P lies on. Conversely, assume two rational points Q and R lie on a circle centered at P. Then the perpendicular bisector of the segment QR passes through P. But this perpendicular bisector must have rational coefficients (since Q and R are both rational), which implies that some relation of the form pπ+qe=r exist

On the other hand, any circle with 3 (or more) rational points on its circumference must be centered at a rational point;

suppose e+π were rational. Then (1,π+e+1), (−1,π+e−1) are rational points and are equidistant from (π,e), which implies there is a circle with center (π,e) that contains two rational points.