QJEEM11Ma1082: JEE Main 2011 Mathematics - Class 11 JEEM11Ma1082 Solution: Let α , β be real and z be a complex number. If z 2 + α z + β = …

Question

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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 :

Accepted Answer

As real part of roots is

1

Let roots are

1 + p i, 1 + q

∴

sum of roots

= 1 + p i + 1 + q i = - α

which is real

\Rightarrow q = - p

or root are

1 + p i

and

1 - p i

product of roots

= 1 + p^{2} = β \in (1, \infty)

p \neq 0

as roots are distinct.

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