Các số phức z thỏa mãn |z-1+2i|=|z+3-i|. Tìm |z|min

* Hướng dẫn giải

Đáp án D

$$\begin{gathered} \mathrm{Đặt} \mathrm{z}=\mathrm{x}+\mathrm{yi} \left(\mathrm{x};\mathrm{y}\in \mathrm{ℝ}\right) \\ \mathrm{Từ} \mathrm{giả} \mathrm{thiết}: \left|\mathrm{z}-1+2\mathrm{i}\right|=\left|\mathrm{z}+3-\mathrm{i}\right| \\ \iff {\left(\mathrm{x}-1\right)}^2+{(\mathrm{y}+2)}^2={\left(\mathrm{x}+3\right)}^2+{(\mathrm{y}-1)}^2\iff -2\mathrm{x}+1+4\mathrm{y}+4=6\mathrm{x}+9-2\mathrm{y}+1 \\ \iff 8\mathrm{x}-6\mathrm{y}+5=0 \\ \Rightarrow \mathrm{Tập} \mathrm{hợp} \mathrm{điểm} \mathrm{M} \mathrm{thỏa} \mathrm{mãn} \mathrm{giả} \mathrm{thiết} \mathrm{nằm} \mathrm{trên} \mathrm{đường} \mathrm{thẳng} \left(\mathrm{d}\right): 8\mathrm{x}-6\mathrm{y}+5=0 \\ \mathrm{Mặt} \mathrm{khác}: \left|\mathrm{z}\right|=\mathrm{OM} \\ {\mathrm{OM}}_{\min }\iff \mathrm{OM}\perp \mathrm{d} \mathrm{hay} {\mathrm{OM}}_{\min }=\mathrm{d}\left(\mathrm{O};\left(\mathrm{d}\right)\right)=\frac{\left|5\right|}{\sqrt{8^2+{(-6)}^2}}=\frac{1}{2} \end{gathered}$$