<單選題>已知\(z_{1}\)，\(z_{2}\)為兩個非零複數，且\(\vert z_{1}+z_{2}\vert=\vert z_{1}-z_{2}\vert\)，則\(\frac{z_{1}}{z_{2}}\)的實部為？
(1)\(0\)
(2)\(\frac{1}{2}\)
(3)\(1\)
(4)\(-\frac{1}{2}\)
(5)\(-1\)
已知\(\vert z_{1}+z_{2}\vert=\vert z_{1}-z_{2}\vert\)，兩邊平方得\((z_{1}+z_{2})(\overline{z_{1}+z_{2}})=(z_{1}-z_{2})(\overline{z_{1}-z_{2}})\)。
即\((z_{1}+z_{2})(\overline{z_{1}}+\overline{z_{2}})=(z_{1}-z_{2})(\overline{z_{1}}-\overline{z_{2}})\)。
展開得\(z_{1}\overline{z_{1}}+z_{1}\overline{z_{2}}+z_{2}\overline{z_{1}}+z_{2}\overline{z_{2}}=z_{1}\overline{z_{1}}-z_{1}\overline{z_{2}}-z_{2}\overline{z_{1}}+z_{2}\overline{z_{2}}\)。
化簡得\(z_{1}\overline{z_{2}}+z_{2}\overline{z_{1}}=0\)。
設\(\frac{z_{1}}{z_{2}}=x+yi\)（\(x,y\in R\)），則\(z_{1}=(x + yi)z_{2}\)，代入\(z_{1}\overline{z_{2}}+z_{2}\overline{z_{1}}=0\)得：
\((x + yi)z_{2}\overline{z_{2}}+z_{2}\overline{(x + yi)z_{2}}=0\)，\((x + yi)\vert z_{2}\vert^{2}+z_{2}\overline{z_{2}}(x - yi)=0\)，\(2x\vert z_{2}\vert^{2}=0\)。
因為\(z_{2}\neq0\)，所以\(x = 0\)，即\(\frac{z_{1}}{z_{2}}\)的實部為\(0\)，答案為(1)。 報錯
ChatGPT DeepSeek
相關試題
