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

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

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

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

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None of these

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Solution

The correct option is B $a c x^{2} + b (a + c) x + {(a + c)}^{2} = 0.$
Given: $α, β$ are roots of $a x^{2} + b x + c = 0$
let $y = α + \frac{1}{β}$
$\Rightarrow y = \frac{α β + 1}{β}$
$\Rightarrow β = \frac{a + c}{a y}$
$a {(\frac{a + c}{a y})}^{2} + b (\frac{a + c}{a y}) + c = 0$
$\Rightarrow \frac{(a + c)^{2}}{a y^{2}} + \frac{b (a + c)}{a y} + c = 0$
$\Rightarrow a c y^{2} + b (a + c) y + (a + c)^{2} = 0$
Hence, option C.

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