Let - - be real and z be a complex number- If z2- z- -0 has two distinct roots on the line Re z - 1- then it is necessary that-

The correct option is D β∈ #160; ( 1,∞ ) z^2+α z+β = 0 The roots are z = α±√(α^2 4β)2 Given roots lie on Re(z) =1 Real part of root ...

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Standard XII

Physics

Introduction

Let α ,β be...

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 $R e (z) = 1$ , then it is necessary that:

β \in (0, 1)

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β \in (- 1, 0)

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| β | = 0

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β \in (1, \infty)

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Solution

The correct option is D $β \in (1, \infty)$
$z^{2} + α z + β = 0$

The roots are $z = \frac{- α \pm \sqrt{α^{2} - 4 β}}{2}$

Given roots lie on $R e (z) = 1$

Real part of root $= - \frac{α}{2} = 1 ⟹ α = - 2$

Roots are distinct and complex.

$∴ α^{2} - 4 β < 0$

$4 - 4 β < 0$

$1 - β < 0$

$- β < - 1$

$∴ β > 1$

$β \in (1, \infty)$

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Let α,β be real and z be a complex number. If z2+αz+β=0 has two distinct roots on the line Re(z)=1, then it is necessary that:

The correct option is D β∈(1,∞)z2+αz+β=0The roots are z=−α±√α2−4β2Given roots lie on Re(z)=1Real part of root =−α2=1⟹α=−2Roots are distinct and complex.∴α2−4β<04−4β<01−β<0−β<−1∴β>1β∈(1,∞)

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 $R e (z) = 1$ , then it is necessary that: