Question

已知ω=z+i(z∈C),是純虛數(shù),又|ω+1|²+|ω-1|²=16,求ω.

Answer 1

解析:設(shè)z=x+yi(z≠0)，?

∴.?

∴.?

∴z·+2z-2-4+z·-2z+2-4=0.?

2|z|²=8,|z|²=4.?

∴x²+y²=4,ω=x+(y+1)i.?

又|ω+1|²+|ω-1|²=16,|(x+1)+(y+1)i|²+|(x-1)+(y+1)i|²=16,?

∴（x+1)²+(y+1)²+(x-1)²+(y+1)²=162x²+2+2y²+4y+2=162x²+2y²+4y-12=0.?

∴2(4-y²)+2y²+4y-12=0,y=1,x=±.?

∴z=±+i，ω=±+2i.