6.238. $\left\{\begin{array}{l}\sqrt{\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}-1\right)^{2}}=1.6, \\ x y=2 .\end{array}\right.$
## Solution.

Domain of definition: $\left\{\begin{array}{l}x \neq 0, \\ y \neq 0 .\end{array}\right.$

We will transform the first equation of the system

$$
\sqrt{\left(\frac{x^{2}-y^{2}-x^{2}-y^{2}}{x^{2}+y^{2}}\right)^{2}}=1.6 \Leftrightarrow \sqrt{\left(\frac{-2 y^{2}}{x^{2}+y^{2}}\right)^{2}}=1.6 \Rightarrow \frac{2 y^{2}}{x^{2}+y^{2}}=1.6
$$

$\frac{y^{2}}{x^{2}+y^{2}}=\frac{4}{5} \Leftrightarrow \frac{x^{2}+y^{2}}{y^{2}}=\frac{5}{4}, \frac{x^{2}}{y^{2}}+1=\frac{5}{4},\left(\frac{x}{y}\right)^{2}=\frac{1}{4} . \quad$ Then $\quad \frac{x}{y}=-\frac{1}{2}$
or $\frac{x}{y}=\frac{1}{2}$ and the given system of equations is equivalent to two systems of equations:

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0387.jpg?height=164&width=731&top_left_y=233&top_left_x=113)

Answer: $(-1 ;-2),(1 ; 2)$.