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  4. 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 (π, e ).

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Q. 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 $(\pi, e )$.

Conic Sections

Solution:

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.
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Hence the maximum number of such points is$2$