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Geometric Progression

lf a,b, g a...

Question

lf $a, b, g$ are the roots of the equation $x^{3} + a x^{2} + b x + c = 0$ , then $a^{- 1} + b^{- 1} + g^{- 1} =$

\frac{a}{c}

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- \frac{b}{c}

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\frac{c}{a}

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\frac{b}{a}

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Solution

The correct option is B $- \frac{b}{c}$
As $a, b, g$ are roots of $x^{3} + a x^{2} + b x + c = 0$
Gives
$S_{1} = a + b + g = - a S_{2} = a b + b g + g a = b S_{3} = a b g = - c$
Therefore $a^{- 1} + b^{- 1} + g^{- 1} = \frac{1}{a} + \frac{1}{b} + \frac{1}{g}$
$= \frac{b g + a b + a g}{a b g} = \frac{b}{- c}$
$= - \frac{b}{c}$

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