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Complex Numbers

If α, β, γ ...

Question

If $α, β, γ$ be the non zero real roots of the equation $x^{3} + p x^{2} + q x + r = 0$ satisfying the relation $α β + 1 = 0$ , then

A

γ = r

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B

r^{2} + p r + q = - 1

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C

\frac{1}{α} + \frac{1}{β} = p + r

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D

Harmonic mean of

α, β, γ = \frac{- 3 r}{q}

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Solution

The correct options are
A $γ = r$
B $\frac{1}{α} + \frac{1}{β} = p + r$
C Harmonic mean of $α, β, γ = \frac{- 3 r}{q}$
D $r^{2} + p r + q = - 1$
$α, β, γ$ be the non zero real roots of the equation $x^{3} + p x^{2} + q x + r = 0$
Then, $α + β + γ = - p$
and $α β γ = - r$
$\Rightarrow γ = r (∵ α β + 1 = 0)$
Also, $α β + (α + β) γ = q$
$\Rightarrow - 1 + (- p - γ) γ = q$
$\Rightarrow - 1 - (p + r) r = q$
$\Rightarrow r^{2} + p r + q = - 1$
$\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β} = \frac{- p - r}{- 1} = p + r$
H.M of $α, β, γ = \frac{3}{\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}}$
$= \frac{3}{p + r + \frac{1}{r}} = \frac{3 r}{p r + r^{2} + 1}$
$= \frac{3 r}{- q} = \frac{- 3 r}{q}$ .

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