Review question

# Can we find this circle's integer and non-integer rational points? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9072

## Question

1. 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 $x^2+y^2=2$.

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 $x^2-y^2=7$.