If $α, β$ are the roots of quadratic equation $a x^{2} + b x + c = 0$ , then ${(a α + b)}^{- 2} + {(a β + b)}^{- 2} =$

\frac{b^{2} - a c}{a^{3} c^{2}}

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\frac{b^{2} - 2 a c}{a^{2} c^{2}}

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\frac{b^{3} - 3 a c}{a^{2} c^{2}}

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None of these

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Solution

The correct option is C $\frac{b^{2} - 2 a c}{a^{2} c^{2}}$
Given $α, β$ are the roots of $a x^{2} + b x + c = 0$
$∴ a α^{2} + b α + c = 0 \Rightarrow a α + b = - \frac{c}{α}$
Similarly $a β + b = - \frac{c}{β}$
$∴ {(a α + b)}^{- 2} + {(a β + b)}^{- 2}$
$= \frac{α^{2} + β^{2}}{c^{2}} = \frac{1}{c^{2}} [{(α + β)}^{2} - 2 α β] = \frac{b^{2} - 2 a c}{a^{2} c^{2}}$

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