## Task B-2.4.

Among the complex numbers that satisfy the condition $|z-4-4i|=\sqrt{2}$, determine the one with the smallest absolute value and the one with the largest absolute value.
## Solution.

$z=x+y i, x, y \in \mathbb{R} . \text{From } |z-4-4 i|=\sqrt{2}$ we get $(x-4)^{2}+(y-4)^{2}=2$.

Let's sketch the set of solutions in the complex plane.

![](https://cdn.mathpix.com/cropped/2024_05_30_964e2f56f6f4ff54223dg-08.jpg?height=520&width=586&top_left_y=1662&top_left_x=655)

The absolute value of the complex number $z=x+y i, x, y \in \mathbb{R}$ represents the distance of the point $Z(x, y)$ from the origin. Therefore, on the given circle, we are looking for the closest and the farthest point from the origin. These points lie on the line $O S$ passing through the origin and the center of the circle, i.e., on the line $y=x$. Let's denote them as $Z_{1}$ and $Z_{2}$. If $T(a, b), a \neq b$ is any point on the circle, by the triangle inequality, we have:

$$
|O S| \leqslant|O T|+|T S| \Leftrightarrow\left|O Z_{1}\right|+r \leqslant|O T|+r \Leftrightarrow\left|O Z_{1}\right| \leqslant|O T| \text {. }
$$

This means that the point $Z_{1}$ is the closest to the origin.

The point $Z_{2}$ has the greatest distance from the origin because $\overline{O Z_{2}}$ is the side opposite the largest angle in the triangle $O Z_{2} T\left(\varangle O T Z_{2}=\varangle O T Z_{1}+90^{\circ}\right)$.

The complex numbers we are looking for are obtained by solving the system of equations

$$
\left\{\begin{array}{l}
(x-4)^{2}+(y-4)^{2}=2 \\
y=x
\end{array}\right.
$$

The solutions to this system are $(3,3)$ and $(5,5)$.

The complex number that satisfies the condition $|z-4-4 i|=\sqrt{2}$ and has the smallest absolute value is $z_{1}=3+3 i$, and the one with the largest absolute value is the number $z_{2}=5+5 i$.

Note: A student can earn one point:

- if they have correctly drawn the circle directly from the condition $|z-4-4 i|=\sqrt{2}$, without its equation.
- if they have written the equation of the circle, even without a sketch.
- if they have both.

Students will intuitively conclude from the drawing that the solutions lie on the line $y=x$. If a student refers to a drawn image showing that they were comparing the distance with another point, and that the smallest and largest distances are indeed at points $Z_{1}$ and $Z_{2}$, there is no need to insist on detailed reasoning. If there is only the line $y=x$ drawn, without any comment, but everything else is correctly calculated, deduct 1 point.