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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

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$β \in (1, \infty)$

$β \in (0, 1)$

$β \in (- 1, 0)$

$| β | = 1$

answer is A.

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Detailed Solution

Let the roots of the given equation be 1 + ip and 1 – ip, where $p \in R$

$\Rightarrow β =$ product of roots

$= (1 + i p) (1 - i p) = 1 + p^{2} > 1, \forall p \in R \Rightarrow β \in (1, \infty)$

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