Question

已知复数z满足：|z|=1+3i-z，求(1+i)2(3+4i)22z的值

已知复数z满足：|z|=1+3i-z，求(1+i)2(3+4i)22z的值．

Answer 1

设z=a+bi，（a，b∈R），而|z|=1+3i-z，
即

a2+b2

?1?3i+a+bi＝0
则 

a2+b2 +a?1＝0 b?3＝0

?

a＝?4 b＝3

，z＝?4+3i

(1+i)2(3+4i)2

2z

＝

2i(?7+24i)

2(?4+3i)

＝

24+7i

4?3i

=

(24+7i)(4+3i)

(4+3i)(4?3i)

=

75+100i

25

=3+4i．