If are the roots of the equation a x 2 + b x + c = 0 , then the value of 1 a + b + 1 a + b equals to

Question

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

Smart Achievers · Accepted Answer

(2)

$Since, α, β are roots of equation$

$a x^{2} + b x + c = 0$

$∴$ $\begin{matrix} α + β = - \frac{b}{a}, & α β = \frac{c}{a}} \end{matrix}$ ...(i)

Taking, $\frac{1}{a α + b} + \frac{1}{a β + b} = \frac{a (α + β) + 2 b}{(a α + b) (a β + b)}$

$= \frac{a (α + β) + 2 b}{a^{2} α β + a b (α + β) + b^{2}}$

$= \frac{a (- \frac{b}{a}) + 2 b}{a^{2} \cdot \frac{c}{a} + a b (- \frac{b}{a}) + b^{2}} = \frac{b}{a c}$ $(Using (i))$

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