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 $
$\Rightarrow h=\pm \sqrt{r^{2}-2}$