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

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Solution

$a + b = - \frac{b}{a}$ and $a b = \frac{c}{a}$
$∴ \frac{1}{a + β} = \frac{a}{b}$ and $\frac{1}{α} + \frac{1}{β} \frac{α + β}{α β} = - \frac{b}{c}$
Equation is $x^{2} - (- \frac{a}{b} - \frac{b}{c}) x + (- \frac{a}{b}) (- \frac{b}{c}) = 0$
$\Rightarrow b c x^{2} + (b^{2} + a c) x + a b = 0$

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