
A3 Find all values of the real parameter $a$, for which the system

$$
\left\{\begin{array}{c}
(|x|+|y|-2)^{2}=1 \\
y=a x+5
\end{array}\right.
$$

has exactly three solutions.


Solution: The first equation is equivalent to

$$
|x|+|y|=1
$$

or

$$
|x|+|y|=3
$$

The graph of the first equation is symmetric with respect to both axes. In the first quadrant it is reduced to $x+y=1$, whose graph is segment connecting points $(1,0)$ and $(0,1)$. Thus, the graph of

$$
|x|+|y|=1
$$

is square with vertices $(1,0),(0,1),(-1,0)$ and $(0,-1)$. Similarly, the graph of

$$
|x|+|y|=3
$$

is a square with vertices $(3,0),(0,3),(-3,0)$ and $(0,-3)$. The graph of the second equation of the system is a straight line with slope $a$ passing through (0,5). This line intersects the graph of the first equation in three points exactly, when passing through one of the points $(1,0)$ or $(-1,0)$. This happens if and only if $a=5$ or $a=-5$.
