Question

12．已知復數(shù)z的共軛復數(shù)為$\overline z$，且z•$\overline z-3iz=\frac{10}{1-3i}$，求復數(shù)z．

Answer 1

分析 設z=a+bi（a，b∈R），則$\overline{z}$=a-bi，代入化簡$\left.\begin{array}{l}{z•\overline{z}-3iz}\end{array}\right.$，再代入z•$\overline z-3iz=\frac{10}{1-3i}$化簡，利用復數(shù)相等的條件列出方程組，求出a、b的值即可求出復數(shù)z．

解答 解：設z=a+bi（a，b∈R），則$\overline{z}$=a-bi，
∴$\left.\begin{array}{l}{z•\overline{z}-3iz={a}^{2}+^{2}+3b-3ai}\end{array}\right.$，
∵z•$\overline z-3iz=\frac{10}{1-3i}$，
∴$\left.\begin{array}{l}{{a}^{2}+^{2}+3b-3ai=\frac{10（1+3i）}{（1-3i）（1+3i）}=1+3i}\end{array}\right.$，
則$\left.\begin{array}{l}{\left\{\begin{array}{l}{{a}^{2}+^{2}+3b=1}\\{-3a=3}\end{array}\right.}\end{array}\right.$，解得$\left.\begin{array}{l}{\left\{\begin{array}{l}{a=-1}\\{b=0}\end{array}\right.}\end{array}\right.$或$\left.\begin{array}{l}{\left\{\begin{array}{l}{a=-1}\\{b=-3}\end{array}\right.}\end{array}\right.$
∴z=-1或$\left.\begin{array}{l}{z=-1-3i}\end{array}\right.$．

點評 本題考查復數(shù)代數(shù)形式的乘除運算，共軛復數(shù)和復數(shù)相等的定義，以及化簡、計算能力．