3.5. Let $\left(x_{0}, y_{0}\right)$ be a solution to the system of equations

$$
\begin{gathered}
-x+a y=2 a \\
a x-y=3 a-5
\end{gathered}
$$

where $a$ is a real parameter. What is the minimum value of the expression $x_{0}^{2}+y_{0}^{2} ?$

For which $a$ is this minimum achieved?

## fourth grade
3.5. For $a \neq \pm 1$ the system has a unique solution
$x=\frac{3 a}{a+1}, y=\frac{2 a+5}{a+1}$.

For $a=-1$ the system has no solutions, and for $a=1$ the solutions are $(x, y) \in\{(\alpha, \alpha+2) \mid \alpha \in R\}$. In the case $a=1$, the minimum value of the expression $x^{2}+y^{2}=\alpha^{2}+(\alpha+2)^{2}$ is 2.3a

For $a \neq \pm 1$ it holds that $\frac{9 a^{2}+(2 a+5)^{2}}{(a+1)^{2}}>2$. The minimum value is 2 and is achieved for $a=1$.

## fourth grade