Question

The question is below?

Let C be any circle with centre `(0,sqrt2)`. Prove that at most two rational points can be there on C

Answer 1

See explanation...

Explanation:

Suppose a circle with centre on the `y` axis passes through more than `2` rational points. Then at least two of those points do not lie on the same horizontal line. Let's call them `A` and `B`.

The midpoint of `AB` is also a rational point `M` and the slope of the line through `M` perpendicular to `AB` is finite and rational.

This line is the set of points equidistant from `A` and `B`, which includes the centre of the circle.

Since `M` is rational and the slope of the line is rational, it intersects the `y` axis at a rational point, which is the centre of the circle.

Since `(0, sqrt(2))` is not rational, there can be no such rational points `A`, `B` on a circle with `(0, sqrt(2))` as its centre.