Z∈C，满足Z+1Z∈R，Z−14是纯虚数，求Z．

Question

Z∈C，满足Z+(1)/(Z)∈R，Z−(1)/(4)是纯虚数，求Z．

Answer 1

设z=a+bi（a∈R，b∈R）
则Z−(1)/(4)=（a-(1)/(4)）+bi
∵Z−(1)/(4)是纯虚数
∴a-(1)/(4)=0，b≠0
∴a=(1)/(4)，b≠0①
∵Z+(1)/(Z)＝a+(a)/(a2+b2)+(b−(b)/(a2+b2))i∈R
∴b−(b)/(a2+b2)=0
∵b≠0
∴a²+b²=1②
∴由①②得b=

+

.

(√(3))/(2)
∴Z=(1)/(4)+(√(3))/(2)i或Z=(1)/(4)-(√(3))/(2)i