设a,b都是虚数,且它们互为共轭复数.巳知a^2/b是实数,求a/b的值

题目

设a,b都是虚数,且它们互为共轭复数.巳知a^2/b是实数,求a/b的值

答案

a,b都是虚数,且它们互为共轭复数
设a=x+yi,那么b=x-yi ,x,y∈R,且y≠0
∴a^2/b
=(x+yi)^2/(x-yi）
=(x+yi)^3/(x^2+y^2)
=(x^3+3x^2yi-3xy^2-y^3i)/(x^2+y^2)
=[(x^3-3xy^2)+(3x^2y-y^3)i]/(x^2+y^2)
∵a^2/b是实数
∴3x^2y-y^3=0
∵y≠0
∴3x^2=y^2
∴y=±√3x
∴a/b
=(x+yi)/(x-yi)
=(x+yi)^2/(x^2+y^2)
=(x^2-y^2+2xyi)/(x^2+y^2)
=(-2x²±2√3x²i)/(-x²)
=2±2√3i