Q. If $\alpha,\ \beta$ are the roots of the equation $ax^{2} + bx + c = 0\ such\ that\ \beta < \alpha < 0,$ then the quadratic equation whose roots are $|\alpha|,\ |\beta|,\$is given by
A) $|\alpha|x^{2} + |b|x + |c| = 0$
B) $ax^{2} - |b|x + c = 0$
C) $|a|x^{2} - |b|x + |c| = 0$
D) $a|x|^{2} + b|x| + c = 0$
Explanation
(c)
Since, $\alpha\ and\ \beta$ be the roots of the equation $ax^{2} + bx + c = 0$, then
$\alpha + \beta = - \frac{b}{a}\ and\ \ \alpha\beta = \frac{c}{a}$
Now, sum of roots $= |\alpha| + |\beta|$
$= - \alpha - \beta$ $(\because\ \beta < \alpha < 0)$
$= - \left( \ - \frac{b}{a} \right) = \left| \frac{b}{a} \right|$ $(\because|\alpha| + |\beta| > 0)$
and product of roots $= |\alpha||\beta| = \left| \frac{b}{a} \right|$
Hence, required equation is
$x^{2} - \left| \frac{b}{a} \right|x + \left| \frac{c}{a} \right| = 0$
$\Rightarrow \ |a|x^{2} - |b|x + |c| = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.