Solution To Mensa Unit Circle Algebra Problem

Ans.  There are infinitely many rational coordinate pairs that satisfy the unit circle equation.

Proof: Let x = a /b, and y = c/d

Also: b  ≠   0,  d  ≠   0

x ²    +    y ²  =  1

=>    x ²    =   1 -     y ² 

 x ²    =   1 -  (c/d) ²   = (d ² - c ²)/ d ²

Since:

x ²    =   (d ² - c ²) / d ²

Then  (d ² - c ²)   must be a perfect square.

For simplicity, let:

 

e ²   =  (d ² - c ²)

è c ²  + e ² =  d ²

Then: c-e-d is a Pythagorean triple.  Because there are an infinite number of Pythagorean triples:

c-e-d (d > c, e)

There are an infinite number of rational coordinate pairs:

( c ²  /d²),   ( e ² /d²)

that satisfy the unit circle equation.