Z∈C,满足Z+1/Z∈R,Z−1/4是纯虚数,求Z.

问题描述:

Z∈C,满足Z+

1
Z
∈R,Z−
1
4
是纯虚数,求Z.

设z=a+bi(a∈R,b∈R)
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