Question

The number of point(s) with rational coordinates on the circumference of a circle having centre $\left(\pi , e\right)$ is

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

Answer 1

Answer: (A)

Let two points $\left(x_{1} , y_{1}\right)$ and $\left(x_{2} , y_{2}\right)$ lie on the circumference of a circle with centre $\left(\pi , e\right)$ where $x_{1},y_{1},x_{2},y_{2}\in Q$
Point $\left(x , y\right)$ equidistant from $\left(x_{1} , y_{1}\right)$ & $\left(x_{2} , y_{2}\right)$ will lie on the perpendicular bisector of this line segment.
$y-\frac{y_{1} + y_{2}}{2}=-\left(\frac{\left(x_{2} - x_{1}\right)}{y_{2} - y_{1}}\right)\left(x - \frac{\left(x_{1} + x_{2}\right)}{2}\right)$
$\Rightarrow px+qy=r=0$ where $p,q,r\in Q$
Now it must pass through $\left(\pi , e\right)\Rightarrow p\pi +qe+r=0$
$\Rightarrow p=q=r=0$
There is no such line possible, hence at most one point lies on the circle.