复数证明10z1^2+5z2^2=2z1z2,z1+2z2为虚数
2009-02-17 20:52:20做***
10z1^2+5z2^2=2z1z2,z1+2z2为虚数,求证:3z1-z2为实数复数证明10z1^2+5z2^2=2z1z2,z1+2z2为虚数,求证:3z1-z2为实数:改为:10z1^2+5z2^2=2z1z2,z1+2z2为纯虚数,求?
最佳回答
改为:10z1^2+5z2^2=2z1z2,z1+2z2为纯虚数,求证:3z1-z2为实数
10z1^2+5z2^2=2z1z2,10z1^2+5z2^2-2z1z2=0,即
(z1^2+4z2^2+4z1z2)+(9z1^2+z2^2-6z1z2)=0
z1+2z2为纯虚数,(z1+2z2)^20,即(3z1-z2)^2>0
∴3z1-z2为实数
2009-02-17 21:35:32
let z1=a+bi,z2=c+di,(a,b,c,d为实数),10z1^2+5z2^2=2z1z2,10(a^2-b^2)+2abi+5(c^2-d^2)+2cdi=2(ac-bd)+2(b+d)i,z1+2z2为虚数,a+2c=0,a=-2c,由复数相等,得d=3b,a=-2c,则3z1-z2=(3a-c)+(3b-d)i=-7c,为实数。
2009-02-18 18:38:57
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