2026/035) Prove that circle l(0,2) with equation $x^2+y^2=4$ contains infinite points with rational coordinates.

 Solution

Solution to this is $x = 2 \sin 2t$ and $y = 2\cos 2t$ (deliberately chosen angle in form of 2t to avoid fraction angle)

We can represent $\sin 2t$ and $\cos 2t$ expressible in form $\tan t$ as

$\sin 2t = \frac{(2 \tan\, t)}{(1+ tan^2 t)}$

$\cos 2t = \frac{(1-tan ^2t)}{(1+ tan^2 t)}$

So we have 

$x = \frac{(4 \tan\, t)}{(1+ tan^2 t)}$

$y = \frac{2(1-tan ^2t)}{(1+ tan^2 t)}$

If $\tan\, t$ is rational then both x and y are rational and point is rational co-ordinate

We can any rational value of $\tan\, t$ to get rational co-ordinate of a point

Hence it has infinite points with rational co-ordinates