1
Question
If z be any complex number and if z2+az+b=0 has roots both of which has unit modulus, then
If z be any complex number and if z2+az+b=0 has roots both of which has unit modulus, then
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Solution
The correct option is C arg(b) = 2 arg (a)
Let z1, z2 be the roots of z2+az+b=0
Let z1=cis α, z2=cis β and
z1+ z2= −a, z1z2=b
∴|b|=1, |z1+ z2|≤ |z1|+|z2| ⟹ |a|≤2
arg (−a) = α + β2 ⟹ π+arg(a)= α+ β2
α+ β=2π+2arg(a)
⟹ α+ β=2arg(a)
⟹arg(b)=2arg(a)
Let z1, z2 be the roots of z2+az+b=0
Let z1=cis α, z2=cis β and
z1+ z2= −a, z1z2=b
∴|b|=1, |z1+ z2|≤ |z1|+|z2| ⟹ |a|≤2
arg (−a) = α + β2 ⟹ π+arg(a)= α+ β2
α+ β=2π+2arg(a)
⟹ α+ β=2arg(a)
⟹arg(b)=2arg(a)
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