Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$ such that $\beta <\alpha <0,$ then the quadratic equation whose roots are $|\alpha |,|\beta |$ is given by

• A. $|\alpha |x^2+|b|x+|c|=0$
• B. $a{x}^2-|b|x+c=0$
• C. $|a|x^2-|b|x+|c|=0$
• D. $a|x{|}^2+b|x|+c=0$

Answer 1

The correct option is C

$|a|x^2-|b|x+|c|=0$

$Sumofroots=|\alpha |+|\beta |=-\alpha -\beta (\because \alpha <0and\beta <0)=-(\alpha +\beta )=-(-\frac{b}{a})=\frac{b}{a}=\left|\frac{b}{a}\right|(\because |\alpha |+|\beta |>0)$
and product of roots $=|\alpha ||\beta |=|\alpha \beta |=\left|\frac{c}{a}\right|$
Hence, required equation is
$x^2-\left(\left|\frac{b}{a}\right|\right)x+\left|\frac{c}{a}\right|=0or|a|x^2-|b|x+|c|=0.$