Can we find this circle's integer and non-integer rational points?

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The point with coordinates \((a,b)\), where \(a\) and \(b\) are rational numbers, is called:

• an integer rational point if both \(a\) and \(b\) are integers;

• a non-integer rational point if neither \(a\) nor \(b\) is an integer.

1. Write down an integer rational point and a non-integer rational point on the circle \(x^2+y^2=1\).

2. Write down an integer rational point on the circle \(x^2+y^2=2\). Simplify \[(\cos\theta+\sqrt{m}\sin\theta)^2+(\sin\theta-\sqrt{m}\cos\theta)^2\] and hence obtain a non-integer rational point on the circle \(x^2+y^2=2\).

The point with coordinates \((p+\sqrt{2}q,r+\sqrt{2}s)\), where \(p\), \(q\), \(r\) and \(s\) are rational numbers, is called:

• an integer \(2\)-rational point if all of \(p\), \(q\), \(r\) and \(s\) are integers;

• a non-integer \(2\)-rational point if none of \(p\), \(q\), \(r\) and \(s\) is an integer.

1. Write down an integer \(2\)-rational point, and obtain a non-integer \(2\)-rational point, on the circle \(x^2+y^2=3\).

2. Obtain a non-integer \(2\)-rational point on the circle \(x^2+y^2=11\).

3. Obtain a non-integer \(2\)-rational point on the hyperbola \(x^2-y^2=7\).