## Task A-4.1.

Determine all complex numbers $z$ for which

$$
|z+1|=|4-\bar{z}| \quad \mathrm{and} \quad \operatorname{Im}\left(\frac{z}{5+i}\right)=\frac{1}{13}
$$
## First Solution.

Let $z=a+bi$. Then we have

$$
\begin{aligned}
& |z+1|=|a+1+bi|=\sqrt{(a+1)^{2}+b^{2}} \\
& |4-\bar{z}|=|3-a+bi|=\sqrt{(4-a)^{2}+b^{2}}
\end{aligned}
$$

By squaring the first equation and substituting the obtained result, we get

$$
\begin{aligned}
(a+1)^{2}+b^{2} & =(4-a)^{2}+b^{2} \\
a^{2}+2a+1 & =16-8a+a^{2} \\
10a & =15 \\
a & =\frac{3}{2}.
\end{aligned}
$$

We can simplify the expression from the second equation first:

$$
\operatorname{Im}\left(\frac{z}{5+i}\right)=\operatorname{Im}\left(\frac{a+bi}{5+i} \cdot \frac{5-i}{5-i}\right)=\operatorname{Im}\left(\frac{5a+b+(5b-a)i}{26}\right)=\frac{5b-a}{26}. \quad 1 \text{ point}
$$

Therefore,

$$
\begin{aligned}
\frac{5b-a}{26} & =\frac{1}{13} \\
5b-a & =2
\end{aligned}
$$

Using $a=\frac{3}{2}$, we get $b=\frac{7}{10}$.

Thus, the only such complex number $z$ is $\frac{3}{2}+\frac{7}{10}i$. 1 point