Question

18．若x+y-30-xyi和60i-|x+yi|是共轭复数，求实数x和y的值．

Answer 1

分析 60i-|x+yi|=60i-$\sqrt{{x}^{2}+{y}^{2}}$．根据x+y-30-xyi和60i-|x+yi|是共轭复数，可得$\left\{\begin{array}{l}{x+y-30=-\sqrt{{x}^{2}+{y}^{2}}}\\{xy=60}\end{array}\right.$，解得即可．

解答 解：60i-|x+yi|=60i-$\sqrt{{x}^{2}+{y}^{2}}$．
∵x+y-30-xyi和60i-|x+yi|是共轭复数，
∴$\left\{\begin{array}{l}{x+y-30=-\sqrt{{x}^{2}+{y}^{2}}}\\{xy=60}\end{array}\right.$，解得：$\left\{\begin{array}{l}{x=5}\\{y=12}\end{array}\right.$或$\left\{\begin{array}{l}{x=12}\\{y=5}\end{array}\right.$．
∴x=5，y=12或x=12，y=5．

点评 本题考查了共轭复数的定义、复数相等、方程组的解法，考查了推理能力与计算能力，属于中档题．