求虚数z,使z+9/z是实数,且|z+2i|=5

设z=x+yi y不等于0
z+9/z=x+yi+9/(x+yi)=x+yi+9(x-yi)/(x²+y²)为实数,所以虚数部分为0
y-9y/(x²+y²)=0
x²+y²=9
|z+2i|=|x+(y+2)i|=5
x²+(y+2)²=25
9-y²+(y+2)²=25
y=3
x=0
z=3i