Task B-2.1. Draw in the complex plane the set of points $(x, y)$, associated with complex numbers $z=x+yi$, for which

$$
\operatorname{Re} z \cdot \operatorname{Im} z<0 \quad \mathrm{and} \quad|z|^{2} \leq \operatorname{Im}\left(z^{2}\right)+1
$$

Calculate the area of this set of points.
## Solution.

From the first inequality, it follows that $x \cdot y < 0$, which means that the desired points lie in the second or fourth quadrant.

From the second inequality, it follows that

$$
x^{2} + y^{2} \leq 2 x y + 1
$$

or $(x-y)^{2} \leq 1$ or $|x-y| \leq 1$.

From this, we have $-1 \leq x-y \leq 1$, or $y \leq x+1$ and $y \geq x-1$.

The set of desired points is a part of the plane between the lines $y=x+1$ and $y=x-1$, and within the second and fourth quadrants, as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_4f4efb97ca3723b253dag-06.jpg?height=919&width=919&top_left_y=1454&top_left_x=576)

From the figure, it is clear that

$$
P=2 \cdot \frac{1 \cdot 1}{2}=1
$$