A. \(\frac{{\sqrt 2 }}{2}.\)
B. \(\sqrt 2\)
C. \(2\sqrt 3\)
D. \(\frac{2}{{\sqrt 3 }}.\)
A
Do \({z_2} \ne 0;\,\,{z_1} + {z_2} \ne 0\) ta có
\(\begin{array}{l} \frac{{{z_1}}}{{{z_1} + {z_2}}} = 1 + \frac{{2{z_1}}}{{{z_2}}} \Leftrightarrow {z_1}.{z_2} + 2{z_1}^2 + {z_2}^2 + 2{z_1}{z_2} \Leftrightarrow 2{z_1}^2 + {z_2}^2 + 2{z_1}.{z_2} = 0\\ \Leftrightarrow 2{\left( {\frac{{{z_1}}}{{{z_2}}}} \right)^2} + 2\frac{{{z_1}}}{{{z_2}}} + 1 = 0 \Leftrightarrow \left[ \begin{array}{l} \frac{{{z_1}}}{{{z_2}}} = - \frac{1}{2} + \frac{1}{2}i\\ \frac{{{z_1}}}{{{z_2}}} = - \frac{1}{2} - \frac{1}{2}i \end{array} \right. \Rightarrow \left| {\frac{{{z_1}}}{{{z_2}}}} \right| = \frac{{\sqrt 2 }}{2} \end{array}\)
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