Có bao nhiêu số phức z thỏa mãn z + z + z 1

Chọn B

Giả sử \(z=x+iy{\rm \; }\left(x,y\in {\rm R}\right).\) Khi đó ta có
\(\left\{\begin{array}{l} {\left|z\right|^{2} =2\left|z+\overline{z}\right|+4} \\ {\left|z-1-i\right|=\left|z-3+3i\right|} \end{array}\right. \)

\(\Leftrightarrow \left\{\begin{array}{l} {x^{2} +y^{2} =4\left|x\right|+4} \\ {\left(x-1\right)^{2} +\left(y-1\right)^{2} =\left(x-3\right)^{2} +\left(y+3\right)^{2} } \end{array}\right.\)
\(\Leftrightarrow \left\{\begin{array}{l} {x^{2} +y^{2} =4\left|x\right|+4} \\ {x-2y=4} \end{array}\right. \Leftrightarrow \left\{\begin{array}{l} {y=\frac{x-4}{2} } \\ {x^{2} +\left(\frac{x-4}{2} \right)^{2} =4\left|x\right|+4} \end{array}\right. \)

\(\begin{array}{l} {\Rightarrow x^{2} +\left(\frac{x-4}{2} \right)^{2} =4\left|x\right|+4} \\ {\Leftrightarrow 5x^{2} -16\left|x\right|-8x=0} \end{array}\)
\(\Leftrightarrow \left[\begin{array}{l} {\left\{\begin{array}{l} {x\ge 0} \\ {5x^{2} -24x=0} \end{array}\right. } \\ {\left\{\begin{array}{l} {x<0} \\ {5x^{2} +8x=0} \end{array}\right. } \end{array}\right. \Leftrightarrow \left\{\begin{array}{l} {x\ge 0} \\ {\left[\begin{array}{l} {x=0} \\ {x=\frac{24}{5} } \end{array}\right. } \end{array}\right. {\rm \; \; }\vee {\rm \; \; }\left\{\begin{array}{l} {x<0} \\ {\left[\begin{array}{l} {x=0} \\ {x=\frac{-8}{5} } \end{array}\right. } \end{array}\right. \)
Do đó có 3số phức \(z=-2i;{\rm \; }z=\frac{24}{5} +\frac{2}{5} i;{\rm \; }z=\frac{-8}{5} -\frac{14}{5} i\)

thỏa mãn yêu cầu bài toán.