Cho số phức z thay đổi, luôn có

C

- Hướng dẫn giải

Đáp án C

Giả sử:

$$\begin{gathered} \text{w}=a+bi\left(a,b\in ℝ\right) \\ \Rightarrow a+bi=\left(1-2i\right)\overline{z}+3i \\ \begin{array}{l}\Rightarrow \overline{z}=\frac{{\displaystyle a+\left(b-3\right)i}}{{\displaystyle 1-2i}}=\frac{{\displaystyle \left[a+\left(b-3\right)i\right]\left(1-2i\right)}}{{\displaystyle 5}}\\ =\frac{{\displaystyle a-2\left(b-3\right)+\left(2a+b-3\right)i}}{{\displaystyle 5}}\\ \Rightarrow \left|\overline{z}\right|=\left|z\right|\\ =\frac{{\displaystyle 1}}{{\displaystyle 5}}\sqrt{{\left[a-2\left(b-3\right)\right]}^2+{\left(2a+b-3\right)}^2}=2\\ \iff {\left(a-2b+6\right)}^2+{\left(2a+b-3\right)}^2=100\\ \iff {\left(a-2b\right)}^2+{\left(2a+b\right)}^2+12\left(a-2b\right)-6\left(2a+b\right)=55\\ \iff 5a^2+5b^2-30b=55\\ \iff a^2+b^2-6b=11\iff a^2+{\left(b-3\right)}^2=20\end{array} \end{gathered}$$