Question

設(shè)z∈C，求滿足z+∈R且|z-2|=2的復(fù)數(shù)z.

Answer 1

z=4或z=±i

方法一 設(shè)z="a+bi" (a,b∈R),
則z+=a+bi+=a+bi+
=a++i∈R.
∴b=.∴b=0或a²+b²=1.
當(dāng)b=0時(shí),z=a,∴|a-2|=2,∴a=0或a=4.
a=0不合題意舍去,∴z=4.
當(dāng)b≠0時(shí),a²+b²="1.                                    " ①
又∵|z-2|=2,∴(a-2)²+b²="4.                                " ②
由①②解得a=,b=±,∴z=±i.
綜上可知,z=4或z=±i.
方法二 ∵z+∈R,∴z+=+,
∴(z-)-=0,(z-)·=0,
∴z=或|z|=1.下同方法一.