Xét các số phức thỏa mãn : |z-2i| = |z-4+i| Tìm |z|min

* Hướng dẫn giải

Đáp án C

$$\begin{gathered} \mathrm{Đặt} \mathrm{z}=\mathrm{x}+\mathrm{yi} (\mathrm{x};\mathrm{y}\in \mathrm{ℝ}) \\ \mathrm{Từ} \mathrm{ĐB}\Rightarrow \left|\mathrm{x}+(\mathrm{y}-2)\mathrm{i}\right|=\left|\mathrm{x}-4+(\mathrm{y}+1)\mathrm{i}\right| \\ \iff {\mathrm{x}}^2+{(\mathrm{y}-2)}^2={\left(\mathrm{x}-4\right)}^2+{(\mathrm{y}+1)}^2 \iff -4\mathrm{y}+4=-8\mathrm{x}+16+2\mathrm{y}+1 \\ \iff 8\mathrm{x}-6\mathrm{y}-13=0 \left(\mathrm{d}\right) \\ \mathrm{Gọi} \mathrm{M}(\mathrm{x};\mathrm{y})\in \left(\mathrm{d}\right) \mathrm{thì}\left|\mathrm{z}\right|=\mathrm{OM} \\ {\left|\mathrm{z}\right|}_{\min }\iff {\mathrm{OM}}_{\min } \end{gathered}$$

Từ hình vẽ ta thấy OM_min khi 

$$\begin{gathered} \mathrm{OM}\perp \mathrm{d} \mathrm{hay} \mathrm{OM}\equiv \mathrm{OH} \\ \mathrm{Mặt} \mathrm{khác} \mathrm{OH}=\mathrm{d}(\mathrm{O};\mathrm{d})=\frac{\left|-13\right|}{\sqrt{8^2+6^2}}=\frac{13}{10} \\ \Rightarrow {\left|\mathrm{z}\right|}_{\min }=\mathrm{OH}=\frac{13}{10} \end{gathered}$$