Question

若z的绝对值等于1,z²＋2z＋1／z为负实数,求复数z

Answer 1

设：z=a+bi

则 a^2+b^2=1

a^2-b^2+2abi+3a+bi为负实数

则虚部为0 2ab-b=0

b=0 a= 1或-1

a=-1/2 b=√3/2或-√3/2

由于是负实数,经检验,只有b=0 a=-1

b=√3/2 a=-1/2

b=-=√3/2a= -1/2

时满足

z=-1或-1/2+√3/2i或 -1/2-√3/2i