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प्रश्न
If α, β are the roots of the equation \[a x^2 + bx + c = 0, \text { then } \frac{1}{a\alpha + b} + \frac{1}{a\beta + b} =\]
विकल्प
c / ab
a / bc
b / ac
none of these.
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उत्तर
b / ac
Given equation:
\[a x^2 + bx + c = 0\]
Also,
\[\alpha\] and \[\beta\] are the roots of the given equation.
Then, sum of the roots = \[\alpha + \beta = - \frac{b}{a}\]
Product of the roots = \[\alpha\beta = \frac{c}{a}\]
\[\therefore \frac{1}{a\alpha + b} + \frac{1}{a\beta + b} = \frac{a\beta + b + a\alpha + b}{(a\alpha + b) (a\beta + b)} \]
\[ = \frac{a(\alpha + \beta) + 2b}{a^2 \alpha\beta + ab\alpha + ab\beta + b^2} \]
\[ = \frac{a(\alpha + \beta) + 2b}{a^2 \alpha\beta + ab\left( \alpha + \beta \right) + b^2}\]
\[ = \frac{a\left( - \frac{b}{a} \right) + 2b}{a^2 \left( \frac{c}{a} \right) + ab\left( - \frac{b}{a} \right) + b^2} \]
\[ = \frac{b}{ac}\]
