Question

If α, β are the roots of the equation \[a x^2 + bx + c = 0, \text { then } \frac{1}{a\alpha + b} + \frac{1}{a\beta + b} =\]

Options

• c / ab

• a / bc

• b / ac

• none of these.

Answer 1

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 = \[\alpha\beta = \frac{c}{a}\]

\[\therefore \frac{1}{a\alpha + b} + \frac{1}{a\beta + b} = \frac{a\beta + b + a\alpha + b}{(a\alpha + b) (a\beta + b)} \]

\[ = \frac{a(\alpha + \beta) + 2b}{a^2 \alpha\beta + ab\alpha + ab\beta + b^2} \]

\[ = \frac{a(\alpha + \beta) + 2b}{a^2 \alpha\beta + ab\left( \alpha + \beta \right) + b^2}\]

\[ = \frac{a\left( - \frac{b}{a} \right) + 2b}{a^2 \left( \frac{c}{a} \right) + ab\left( - \frac{b}{a} \right) + b^2} \]

\[ = \frac{b}{ac}\]