# Find the complex number that satisfy two conditions

Find the complex number $$z$$, that satisfy two conditions;

$$i \cdot Re(\frac{\overline{z}}{z_{1}})=\frac{2-3\sqrt{3}}{13}$$; where

$$Im(\frac{z\cdot z_{1}}{2})=\frac{-3-2\sqrt{2}}{2}$$

where $$z_{1}=2-3i$$.

This problem was given at college mathematics mid-term exam. What is the correct approach for solving the problem, and is it even possible to do so?How can imaginary unit multiplied by a real part ever equal to a real number?

• You are right, the left side is in $\Bbb R\cdot i$, and the right side is in $\Bbb R^{\times}$, which is a contradiction - or just a typo, i.e, $i\cdot$ means $(i)$ as an index. Another possibility is, that there is simply no such number $z$. Commented Dec 8, 2022 at 19:20
• If you remove the $i$ in the first equation, there is exactly one solution in ${\mathbb C}$ Commented Dec 8, 2022 at 19:28
• @DietrichBurde No, it doesn't, the question was given exactly as I wrote it. Commented Dec 8, 2022 at 19:32
• @mortex Yes, I believe you, this is why I made the comment. It might convince you that the $i$ is a typo, because then everything makes sense. If not, the question is trivial, as you have already noted. Then there is no such $z$. Commented Dec 8, 2022 at 19:44