｜z1｜=5,z2=3+4i,且z1乘Z2为纯虚数,求z1

Question

｜z1｜=5,z2=3+4i,且z1乘Z2为纯虚数,求z1

Answer 1

设z1=a+bi,则√（a^2+b^2)=5,
a^2+b^2=25,(1)
z1*z2=(a+bi)(3+4i)=3a+4ai+3bi-4b=(3a-4b)+(4a+3b)i,
∵乘积为纯虚数,则实部为0,且虚部非0,
3a-4b=0,a=4b/3,(2)
代入（1）式,
b1=3,b2=-3
a1=4,a2=-4,经验算,虚部不为0,
∴z1=4+3i,或z1=-4-3i.