# Problem 11.5 (7 points)

Solve the system of equations $\left\{\begin{array}{l}x^{2}+y^{2}=34 \\ x-y+\sqrt{\frac{x-y}{x+y}}=\frac{20}{x+y}\end{array}\right.$

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# Solution:

Consider two cases:

1) Let $x+y>0$, then $x-y \geq 0 ; \sqrt{(x+y)^{2}}=x+y$.

By eliminating the denominator, solve the second equation of the system, $x^{2}-y^{2}+\sqrt{x^{2}-y^{2}}=20$. Let $\sqrt{x^{2}-y^{2}}=t, t \geq 0, t^{2}+t=20 \Rightarrow t=-5, t=4$. Find the solutions of the system $\left\{\begin{array}{l}x^{2}-y^{2}=16 \\ x^{2}+y^{2}=34\end{array} \Rightarrow 2 x^{2}=50, x= \pm 5, y= \pm 3\right.$ (4 roots). The roots that satisfy the conditions are $(5 ;-3)$ and $(5 ; 3)$.

2) $x+y<0$, then $x-y<0 ; \sqrt{(x+y)^{2}}=-(x+y)$. Solve the second equation of the system $x^{2}-y^{2}-\sqrt{x^{2}-y^{2}}=20$,

$\sqrt{x^{2}-y^{2}}=5, \sqrt{x^{2}-y^{2}}=-4 \Rightarrow\left\{\begin{array}{l}x^{2}-y^{2}=25 \\ x^{2}+y^{2}=34\end{array} \Rightarrow 2 x^{2}=59, x= \pm \frac{\sqrt{118}}{2}, y= \pm \frac{3 \sqrt{2}}{2} . \quad\right.$ The roots that satisfy the conditions are: $\left(-\frac{\sqrt{118}}{2} ; \frac{3 \sqrt{2}}{2}\right),\left(-\frac{\sqrt{118}}{2} ;-\frac{3 \sqrt{2}}{2}\right)$.

| Criteria | Points |
| :--- | :---: |
| Complete solution | 7 |
| Correct approach with arithmetic error | 5 |
| Considering one case | 3 |

Answer: (5;-3), (5;3), $\left(-\frac{\sqrt{118}}{2} ; \frac{3 \sqrt{2}}{2}\right),\left(-\frac{\sqrt{118}}{2} ;-\frac{3 \sqrt{2}}{2}\right)$.