Question

Let $S$ is a circle with centre $(0,\sqrt{2})$. Then
[A rational point is a point both of whose coordinates are rational numbers

• A. There cannot be any rational point on $S$
• B. There can be infinitely many rational points on $S$
• C. There can be at most two rational points on $S$
• D. There are exactly two rational points on $S$

Answer 1

Answer: (C)

The equation of the circle $S$ is $x^2+(y-\sqrt{2}{)}^2=r^2....(1)$
Let the coordinates of any point on this circle be $(h,k)$, then
$h^2+(k-\sqrt{2}{)}^2=r^2$
$\Rightarrow k=\sqrt{2}\pm \sqrt{r^2-h^2}...(2)$
Since the above value of k contains a constant irrational number $\sqrt{2}$, therefore, the only possible rational value of $k$ is 0 .
Hence, $\sqrt{2}\pm \sqrt{r^2-h^2}=0$
$\Rightarrow r^2-h^2=2$
$\Rightarrow h=\pm \sqrt{r^2-2}$