Question 144
The real numbers $a$, $x$, $y$ are such that $a \gt 0$, $x \neq y$ and $x + \sqrt{a - x^2} = y + \sqrt{a - y^2}$ (C).
a) Prove that, if $x$ and $y$ are rational, then $a$ is rational.
b) Prove that, if $a$ is a positive integer, then there exists a pair $(x, y)$, with $x$ and $y$ irrational, fulfilling conditions (C).
a) Prove that, if $x$ and $y$ are rational, then $a$ is rational.
b) Prove that, if $a$ is a positive integer, then there exists a pair $(x, y)$, with $x$ and $y$ irrational, fulfilling conditions (C).