Question

设z∈C，求满足z+∈R且|z-2|=2的复数z.

Answer 1

z=4或z=±i

解析:

方法一  设z=a+bi (a,b∈R),

则z+=a+bi+=a+bi+

=a++i∈R.

∴b=.∴b=0或a²+b²=1.

当b=0时,z=a,∴|a-2|=2,∴a=0或a=4.

a=0不合题意舍去,∴z=4.

当b≠0时,a²+b²=1.                                                                                            ①

又∵|z-2|=2,∴(a-2)²+b²=4.                                                                              ②

由①②解得a=,b=±,∴z=±i.

综上可知,z=4或z=±i.

方法二  ∵z+∈R,∴z+=+,

∴(z-)-=0,(z-)·=0,

∴z=或|z|=1.下同方法一.