Xét các số phức z, w thỏa mãn $\left| z \right|=2$ và $\left| w...

Ta có $\left| w \right|=3\Rightarrow \left| i\bar{w} \right|=3$ ; $\left| z+i\bar{w} \right|\le \left| z \right|+\left| i\bar{w} \right|=2+3=5$.
Khi đó $P=\left| z+i\bar{w}+12+5i \right|\ge \left| 12+5i \right|-\left| z+i\bar{w} \right|\ge 13-5=8$
Suy ra ${{P}_{\min }}=8$ khi $\left\{ \begin{aligned}
& z=k.i\bar{w},\left( k\ge 0 \right) \\
& 12+5i=h.\left( z+i\bar{w} \right),\left( h\le 0 \right) \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& k=\dfrac{2}{3} \\
& h=-\dfrac{13}{5} \\
& z=\dfrac{-24}{13}-\dfrac{10}{13}i \\
& \bar{w}=\dfrac{-15}{13}+\dfrac{36}{13}i \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& z=\dfrac{-24}{13}-\dfrac{10}{13}i \\
& w=\dfrac{-15}{13}-\dfrac{36}{13}i \\
\end{aligned} \right.$
Vậy $\left| z-2w-8 \right|=\left| \dfrac{-24}{13}-\dfrac{10}{13}i-2\left( \dfrac{-15}{13}-\dfrac{36}{13}i \right)-8 \right|=\left| \dfrac{-98}{13}+\dfrac{62}{13}i \right|=\dfrac{82\sqrt{2}}{13}$