If $α, β$ are the roots of quadratic equation $a x^{2} + b x + c = 0$ , then the quadratic equation whose roots are $\frac{1}{a α + b}, \frac{1}{a β + b}$ , is

a c x^{2} - b x + 1 = 0.

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a c x^{2} - 2 b x - 1 = 0.

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2 a c x^{2} + b x + 1 = 0.

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None of these

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Solution

The correct option is A $a c x^{2} - b x + 1 = 0.$
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 $a β + b = - \frac{c}{β}$
Thus roots are $- \frac{α}{c}, - \frac{β}{c}$
$∴ S = - \frac{α + β}{c} = \frac{b}{a c}, P = \frac{α β}{c^{2}} = \frac{1}{a c}$
Hence required quadratic is, $x^{2} - S x + p = 0$
$∴ a c x^{2} - b x + 1 = 0$

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