2. Determine $x, y \in \mathbb{Q}$ that satisfy $\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}}=-11 x+2 y$ and $\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}=2 x+11 y$. GM

## Scoring Guide:

Consider $z=x+i y, x, y \in \mathbb{R}$. We observe that $z^{-1}=\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{y}{\sqrt{x^{2}+y^{2}}} i \quad \ldots 1 p$ $z^{-2}=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}-\frac{2 x y}{x^{2}+y^{2}} i$

Multiply the second equation by $(-i)$ and add it to the first, then rewrite the obtained equation using $z$.

$\frac{x^{2}-y^{2}}{x^{2}+y^{2}}-\frac{2 x y}{x^{2}+y^{2}} i=-11 x+2 y-2 x i-11 y i$

$z^{-1}=-(11+2 i) z \Rightarrow z^{3}=-\frac{1}{11+2 i} \Rightarrow z^{3}=\left(\frac{1-2 i}{5}\right)^{3}$

Rational solutions are obtained only for $z=\frac{1-2 i}{5}, \quad x=\frac{1}{5}, y=-\frac{2}{5}$
2. Determine $x, y \in \mathbb{Q}$ that satisfy $\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}}=-11 x+2 y$ and $\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}=2 x+11 y$.