Q. In writing an equation of the form $ax^{2} + bx + c = 0;$ the coefficient of $x$ is written incorrectly and roots are found to be equal. Again in writing the same equation the constant term is written incorrectly and it is found that one root is equal to those of the previous wrong equation while the other is double of it. If $\alpha\ and\ \beta$ be the roots of correct equation, then $(\alpha - \beta)^{2}$ is equal to
A) 5
B) 5 $\alpha\ \beta$
C) $- 4\ \alpha\ \beta$
D) $- 4$
Explanation
(b)
Let the correct equation is
$ax^{2} + bx + c = 0$,
Then $\alpha + \beta = - \frac{b}{a}\ and\ \alpha\beta = \frac{c}{a}$
When $b$ is written incorrectly, then the roots are equal.
Let these are $\gamma$ and $\gamma$.
$\therefore\ \gamma.\gamma = \frac{c}{a} \Rightarrow \ \gamma^{2} = \alpha\beta$ …(i)
When $c$ is written icorrectly, then the roots are $\gamma$ and 2$\gamma$.
$\therefore\ \gamma + 2\gamma = - \frac{b}{a}\ \Rightarrow 3\gamma = \alpha + \beta$
$\Rightarrow 9\gamma^{2} = (\alpha + \beta)^{2}\ \Rightarrow 9\alpha\beta = (\alpha - \beta)^{2} + 4\alpha\beta$ [from Eq. (i)]
$\therefore(\alpha - \beta)^{2} = 5\alpha\beta$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.