Let α , β be real and z be a complex number. If z 2 + α z + β = 0 has two distinct roots on the line Re z = 1 , then it is necessary that

Question

Let α , β be real and z be a complex number. If z 2 + α z + β = 0 has two distinct roots on the line Re z = 1 , then it is necessary that, β ∈ ( − 1 , 0 ), ∣ β ∣ = 1, β ∈ ( 1 , ∞ ), β ∈ ( 0 , 1 )

MARKS App · Accepted Answer

Suppose roots are 1 + pi , 1 + qi Sum of roots 1 + p i + 1 + q i = − α which is real ⇒ roots of 1 + pi , 1 − pi Product of roots = β = 1 + p 2 ∈ ( 1 , ∞ ) p  = 0 since roots are distinct.

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