Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$, then $\frac{\alpha }{a\beta +b}+\frac{\beta }{a\alpha +b}=$

• A. $\frac{2}{a}$
• B. $\frac{2}{b}$
• C. $\frac{2}{c}$
• D. $-\frac{2}{a}$

Answer 1

The correct option is D $-\frac{2}{a}$

Given $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$
$\Rightarrow a{\alpha }^2+b\alpha =-c$
$\Rightarrow a\alpha +b=\frac{-c}{\alpha }$ ....(1)
Also, $a{\beta }^2+b\beta =-c$
$\Rightarrow a\beta +b=\frac{-c}{\beta }$ ....(2)
Consider,$\frac{\alpha }{a\beta +b}+\frac{\beta }{a\alpha +b}$
$=\frac{\alpha }{-\frac{c}{\beta }}+\frac{\beta }{-\frac{c}{\alpha }}$
$=\frac{\alpha \beta }{-c}+\frac{\alpha \beta }{-c}$
$=\frac{-2\alpha \beta }{c}=\frac{-2}{c}\times \frac{c}{a}$
$=-\frac{2}{a}$