## Task B-4.7.

In the complex plane, represent the set of all complex numbers $z$ for which $|z| \leqslant 4$ and $2 \sqrt{3}-2 \leqslant \operatorname{Re}(z)+\operatorname{Im}(z) \leqslant 4$. What is the area of this set?
## Solution.

Let $z=x+y i$. Then from $|z| \leqslant 4$ it follows that $x^{2}+y^{2} \leqslant 16$. The points $(x, y)$ for which the given inequality holds form a circle centered at the origin with a radius of 4.

From $2 \sqrt{3}-2 \leqslant \operatorname{Re}(z)+\operatorname{Im}(z) \leqslant 4$ it follows that $2 \sqrt{3}-2 \leqslant x+y \leqslant 4$, which describes the points $(x, y)$ that lie between the lines $y=2 \sqrt{3}-2-x$ and $y=4-x$, or on them.

It follows that the given set of complex numbers is represented by the shaded figure in the image.

![](https://cdn.mathpix.com/cropped/2024_05_30_d4daff78382705a7c34eg-28.jpg?height=643&width=666&top_left_y=1940&top_left_x=181)

The points $C$ and $D$ are the intersections of the circle $x^{2}+y^{2}=16$ and the line $x+y=4$, so they are $C(0,4)$ and $D(4,0)$.

The points $A$ and $B$ are obtained as the intersections of the circle and the second line, or as the solutions of the system $x^{2}+y^{2}=16, x+y=2 \sqrt{3}-2$. It follows that $A(2 \sqrt{3},-2)$ and $B(-2,2 \sqrt{3})$.

Let $M$ be the perpendicular dropped from $B$ to the $x$-axis. In the right triangle $M O B$, $|O M|=2$ and $|B O|=4$, so $\cos \angle M O B=\frac{1}{2}$, or $\angle M O B=60^{\circ}$. It follows that $\angle C O B=30^{\circ}$, or $\angle A O B=150^{\circ}=\frac{5 \pi}{6}$.

The area of the shaded figure in the image can be obtained by subtracting the area of triangle $B O A$ and the area of the circular segment $C D$ from the area of the circular sector $A O B$. It follows that the desired area is

$$
\begin{aligned}
& P=\frac{16 \pi \cdot \frac{5 \pi}{6}}{2 \pi}-\frac{1}{2} \cdot 4 \cdot 4 \cdot \sin \frac{5 \pi}{6}-\left(\frac{1}{4} \cdot 16 \pi-\frac{1}{2} \cdot 16\right) \\
& =\frac{8 \pi}{3}+4
\end{aligned}
$$

Note: If a student has not stated that it is a circle and lines before sketching, but it is visible from the sketch that they have recognized them, they receive all 3 points for that part.

If a student has not shaded the desired figure on the sketch, they can receive a maximum of 2 (out of possible 3) points for that part of the problem.