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.