Google
Câu hỏi: Cho số phức ${{z}_{1}},{{z}_{2}}$ thỏa mãn $\left| {{z}_{1}}+{{z}_{2}} \right|=3,\left| {{z}_{1}} \right|=1,\left| {{z}_{2}} \right|=2.$ Tính ${{z}_{1}}.\overline{{{z}_{2}}}+\overline{{{z}_{1}}}.{{z}_{2}}$
A. 2
B. 8
C. 0
D. 4
A. 2
B. 8
C. 0
D. 4
HD: Ta có $\left| {{z}_{1}}+{{z}_{2}} \right|=3\Leftrightarrow {{\left| {{z}_{1}}+{{z}_{2}} \right|}^{2}}=9\Leftrightarrow \left( {{z}_{1}}+{{z}_{2}} \right)\left( \overline{{{z}_{1}}}+\overline{{{z}_{2}}} \right)=9$
$\Leftrightarrow {{z}_{1}}.\overline{{{z}_{1}}}+{{z}_{1}}.\overline{{{z}_{2}}}+{{z}_{2}}.\overline{{{z}_{1}}}+{{z}_{2}}.\overline{{{z}_{2}}}=9\Leftrightarrow {{\left| {{z}_{1}} \right|}^{2}}+{{z}_{1}}.\overline{{{z}_{2}}}+{{z}_{2}}.\overline{{{z}_{1}}}+{{\left| {{z}_{2}} \right|}^{2}}=9\Rightarrow {{z}_{1}}.\overline{{{z}_{2}}}+{{z}_{2}}.\overline{{{z}_{1}}}=4.$
$\Leftrightarrow {{z}_{1}}.\overline{{{z}_{1}}}+{{z}_{1}}.\overline{{{z}_{2}}}+{{z}_{2}}.\overline{{{z}_{1}}}+{{z}_{2}}.\overline{{{z}_{2}}}=9\Leftrightarrow {{\left| {{z}_{1}} \right|}^{2}}+{{z}_{1}}.\overline{{{z}_{2}}}+{{z}_{2}}.\overline{{{z}_{1}}}+{{\left| {{z}_{2}} \right|}^{2}}=9\Rightarrow {{z}_{1}}.\overline{{{z}_{2}}}+{{z}_{2}}.\overline{{{z}_{1}}}=4.$
Đáp án D.