Task 8. I variant.

To plot on the coordinate plane $O a b$ the set of points for which the system

$$
\left\{\begin{array}{l}
x^{2}+y^{2}=a^{2} \\
|x|+|y|=|b|
\end{array}\right.
$$

has at least one solution
Solution. The graph of the first equation is a circle of radius $|a|$ centered at the origin; the graph of the second equation is a square with vertices on the coordinate axes, half the diagonal of which is $|b|$ (see figure). The system has a solution if the condition $|a| \leq|b| \leq \sqrt{2}|a|$ is satisfied.

![](https://cdn.mathpix.com/cropped/2024_05_06_1cdabbf1f496496d2444g-06.jpg?height=622&width=680&top_left_y=2299&top_left_x=1282)

The set of points in the plane $O a b$ that satisfy this condition is shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_1cdabbf1f496496d2444g-07.jpg?height=654&width=691&top_left_y=193&top_left_x=177)