4. Find all values of the parameter $a$ for which the system

$$
\left\{\begin{array}{c}
(a y-a x+2)(4 y-3|x-a|-x+5 a)=0 \\
\left(\log _{a} x^{2}+\log _{a} y^{2}-2\right) \log _{2} a^{2}=8
\end{array}\right.
$$

has six distinct solutions. (16 points)
Solution. Simplify the second equation of the system: $\left(\log _{a} x^{2}+\log _{a} y^{2}-2\right) \log _{2} a^{2}=8 \Leftrightarrow a>$ $0, a \neq 1, \log _{a} x^{2} y^{2}=2+4 \log _{a} 2,|x y|=4 a$. We have

![](https://cdn.mathpix.com/cropped/2024_05_06_a494f1af39843ff219fbg-04.jpg?height=229&width=1208&top_left_y=1630&top_left_x=184)

I. $\left\{\begin{array}{c}y=x-2 / a \\ |x y|=4 a, a>0, a \neq 1 \text {. }\end{array}\right.$

1) The system has 2 different solutions if

$$
\frac{2}{a}<4 \sqrt{a}, 0<a<\frac{1}{\sqrt[3]{4}}, a \neq 1
$$

Find these solutions:

$x-\frac{2}{a}=\frac{4 a}{x}, x^{2}-\frac{2}{a} x-4 a=0$,

$x_{1 / 2}=\frac{1 \pm \sqrt{1+4 a^{3}}}{a}, y_{1 / 2}=\frac{-1 \pm \sqrt{1+4 a^{3}}}{a}$.

2) The system has 3 different solutions if

$$
\frac{2}{a}=4 \sqrt{a}, \quad a=\frac{1}{\sqrt[3]{4}}
$$

Find these solutions:

$$
\begin{gathered}
x_{1 / 2}=\frac{1 \pm \sqrt{1+4 a^{3}}}{a}, y_{1 / 2}=\frac{-1 \pm \sqrt{1+4 a^{3}}}{a} \\
x_{3}=2 \sqrt{a}, y_{3}=-2 \sqrt{a}
\end{gathered}
$$

3) The system has 4 different solutions if

$$
\frac{2}{a}>4 \sqrt{a}, 0<a<\frac{1}{\sqrt[3]{4}}, a \neq 1
$$

II. $\left\{\begin{array}{c}y=3|x-a| / 4+x / 4-5 a / 4, a>0, a \neq 1\end{array}\right.$ $y=3|x-a| / 4+x / 4-5 a / 4$, for $x \geq a$ we have $y=x-2 a$, for $x \leq$ $a$ we have $y=-\frac{x+a}{2}$.

1) The system has 2 different solutions if

$$
2 a<4 \sqrt{a}, 0<a<4
$$

2) The system has 3 different solutions if

$$
2 a=4 \sqrt{a}, a=4
$$

3) The system has 4 different solutions if

$$
2 a>4 \sqrt{a}, a>4
$$

4) The system has 6 different solutions if

$$
2 a>4 \sqrt{a}, a>32
$$

Find these solutions:

$$
\begin{aligned}
& x_{1}=a+\sqrt{a^{2}+4 a}, y_{1}=-a+\sqrt{a^{2}+4 a} ; x_{2 / 3}=\frac{-a \pm \sqrt{a^{2}+32 a}}{2}, y_{2 / 3}=\frac{-a \mp \sqrt{a^{2}+32 a}}{4} \\
& x_{4}=a+\sqrt{a^{2}-4 a}, y_{4}=-a+\sqrt{a^{2}-4 a}, x_{5 / 6}=\frac{-a \pm \sqrt{a^{2}-32 a}}{2}, y_{5 / 6}=\frac{-a \mp \sqrt{a^{2}-32 a}}{4} .
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_a494f1af39843ff219fbg-06.jpg?height=474&width=1608&top_left_y=1982&top_left_x=201)

The following cases of solution coincidence in I and II cases are possible:

1) $x-\frac{2}{a}=x-2 a, a=1$, in this case there are no solutions;
2) the lines $y=x-\frac{2}{a}, y=-\frac{x+a}{2}$ and the hyperbola $y=\frac{4 a}{x}$ intersect at one point, but this case is possible for $a>32$, and in this case there will be 7 solutions.

Answer: $a \in(0 ; 1 / \sqrt[3]{4}) \cup(4 ; 32)$,