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+
∈R,Z−1 Z
是纯虚数,求Z. 1 4
答
设z=a+bi(a∈R,b∈R)
则Z−
=(a-1 4
)+bi1 4
∵Z−
是纯虚数1 4
∴a-
=0,b≠01 4
∴a=
,b≠0①1 4
∵Z+
=a+1 Z
+(b−a
a2+b2
)i∈Rb
a2+b2
∴b−
=0b
a2+b2
∵b≠0
∴a2+b2=1②
∴由①②得b=
+ .
3
2
∴Z=
+1 4
i或Z=
3
2
-1 4
i
3
2