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Câu hỏi: Cho các số phức ${{z}_{1}},{{z}_{2}}$ thỏa mãn $\left| {{z}_{1}}+{{z}_{2}}+i \right|=1$ và $\left| 3{{z}_{1}}-{{z}_{2}} \right|=10$. Khi đó $P=\left| 4{{z}_{2}}+5+3i \right|$ đạt giá trị nhỏ nhất thì $\left| {{z}_{1}}+2{{z}_{2}} \right|$ bằng
A. $\dfrac{\sqrt{57}}{4}$.
B. $\dfrac{\sqrt{55}}{4}$.
C. $\dfrac{\sqrt{58}}{4}$.
D. $\dfrac{\sqrt{14}}{2}$.
A. $\dfrac{\sqrt{57}}{4}$.
B. $\dfrac{\sqrt{55}}{4}$.
C. $\dfrac{\sqrt{58}}{4}$.
D. $\dfrac{\sqrt{14}}{2}$.
Đặt $\left\{ \begin{aligned}
& u={{z}_{1}}+{{z}_{2}} \\
& v=3{{z}_{1}}-{{z}_{2}} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& {{z}_{1}}=\dfrac{u+v}{4} \\
& {{z}_{2}}=\dfrac{3u-v}{4} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \left| u+i \right|=1 \\
& \left| v \right|=10 \\
\end{aligned} \right.$.
$\Rightarrow P=\left| 4{{z}_{2}}+5+3i \right|=\left| 3u-v+5+3i \right|=\left| 3(u+i)-v+5 \right|$.
$\ge \left| \left| 3(u+i) \right|-\left| -v+5 \right| \right|=\left| 3-\left| -v+5 \right| \right|$.
Mặt khác: $\left| \left| -v \right|-5 \right|\le \left| -v+5 \right|\le \left| -v \right|+5$.
$\Rightarrow P\ge \left| 3-\left| -v+5 \right| \right|\ge \left| 3-5 \right|=2$.
Dấu "=" xảy ra khi $\left\{ \begin{aligned}
& v=10 \\
& u+i=1 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& v=10 \\
& u=1-i \\
\end{aligned} \right.$$\Rightarrow \left| {{z}_{1}}+2{{z}_{2}} \right|=\left| \dfrac{u+v}{4}+2\dfrac{3u-v}{4} \right|=\left| \dfrac{7u-v}{4} \right|=\dfrac{\sqrt{58}}{4}$.
& u={{z}_{1}}+{{z}_{2}} \\
& v=3{{z}_{1}}-{{z}_{2}} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& {{z}_{1}}=\dfrac{u+v}{4} \\
& {{z}_{2}}=\dfrac{3u-v}{4} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \left| u+i \right|=1 \\
& \left| v \right|=10 \\
\end{aligned} \right.$.
$\Rightarrow P=\left| 4{{z}_{2}}+5+3i \right|=\left| 3u-v+5+3i \right|=\left| 3(u+i)-v+5 \right|$.
$\ge \left| \left| 3(u+i) \right|-\left| -v+5 \right| \right|=\left| 3-\left| -v+5 \right| \right|$.
Mặt khác: $\left| \left| -v \right|-5 \right|\le \left| -v+5 \right|\le \left| -v \right|+5$.
$\Rightarrow P\ge \left| 3-\left| -v+5 \right| \right|\ge \left| 3-5 \right|=2$.
Dấu "=" xảy ra khi $\left\{ \begin{aligned}
& v=10 \\
& u+i=1 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& v=10 \\
& u=1-i \\
\end{aligned} \right.$$\Rightarrow \left| {{z}_{1}}+2{{z}_{2}} \right|=\left| \dfrac{u+v}{4}+2\dfrac{3u-v}{4} \right|=\left| \dfrac{7u-v}{4} \right|=\dfrac{\sqrt{58}}{4}$.
Đáp án C.