IIT JEE , AIEEE Forum - Mathematics , Algebra

konichiwa2x

thanks! I should have spent more time on it.

And for your 2nd question, i.e the condition that a circle will not contain any rational point on its circumference is that both x-coordinate and y-coordinate of the centre of the circle must be irrational and such a circle should not pass through the origin.

Let us assume all the points on the circle are irrational.
The family of all circles with centre

is

Let

and

be two such rational points on the circumferrence of the circle.
Hence,

Quite clearly, this is not possible for possible for any rational

except

.

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