Mensa Intermediate Algebra Problem Based On The Unit Circle

 

The equation for the unit circle in Cartesian coordinates is:

x ²    +    y ²  =  1

Consider the coordinate pairs that satisfy the unit circle equation AND both x and y are rational. That is, they can be expressed as a quotient of two integers, e.g. (-1, 0), (0, 1), (0, -1) and (1, 0).

Are there a finite number of such ordered pairs. or infinitely many?  Prove your answer.