Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^{2}+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re} z=1$, then it is necessary that

JEE Main · 2011 · 10 Apr 2011 · Offline · Complex Numbers

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

Answer: (C)

If the two distinct roots lie on

Re z = 1

, write them as

1 + p i

and

1 + q i

. Their sum

2 + (p + q) i = - α

must be real, so

p + q = 0

and the roots are the conjugate pair

1 \pm p i

. Their product is

β = (1 + p i) (1 - p i) = 1 + p^{2}

. Distinct roots require

p \neq = 0

, so

p^{2} > 0

and

β = 1 + p^{2} > 1

, that is

β \in (1, \infty)

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option