z1 and z2 are the roots of the equation z2−az+b=0, where |z1|=|z2|=1 and a, b are non zero complex numbers, then
A
|a|≤1
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B
|a|≤2
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C
arg(a2)=arg(b)
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D
arga=arg(b2)
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Solution
The correct options are A|a|≤2 Barg(a2)=arg(b) z1 and z2 are the roots of the equation z2−az+b=0. Hence, z1+z2=a,z1z2=b Now, |z1+z2|≤|z1|+|z2| ⇒|z1+z1|=|a|≤1+1=2(∵|z1|=|z2|=1) ⇒arg(a)=12[arg(z2)+arg(z1)] Also, arg(b)=arg(z1z2)=arg(z1)+arg(z2) ⇒2arg(a)=arg(b) ⇒arg(a2)=arg(b)