The equation of the family of circles with radius (0,2) is

x²+(y-2)² = r².

 

It is easily proved that if r is rational then there are no rational points.

 

Now, with r irrational, let's assume that we have at least one rational point (x₁,y₁) on the circle. Now, suppose (x₂,y₂) is another such point, we must have

 

x₁²+(y₁-2)² = x₂²+(y₂-2)².

Simplifying we get

(x₁²+y₁²-x₂²-y₂²) + 22(y₂-y₁) = 0.

 

we know that if a+b2 = 0 with a,b rational then a=0 and b= 0.

 

This gives y₁ = y₂. Correspondingly x₂ = -x₁. Hence, there is only one other rational point which is (-x₁,y₁). Hence, if rational points exist there are atmost 2 such points.