Question

Let $\alpha ,\beta ,\gamma$ be the roots of the equation $x^3+3a{x}^2+3bx+c=0.$ If $\alpha ,\beta ,\gamma$ are in H.P., then $\beta$ is equal to

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

Answer 1

The correct option is B $-\frac{c}{b}$

Given $\alpha ,\beta ,\gamma$ are the roots of the equation

$x^3+3a{x}^2+3bx+c=0$

$\Rightarrow \alpha +\beta +\gamma =-3a$

$\alpha \beta \gamma =-c$

$\alpha \beta +\beta \gamma +\gamma \alpha =3b$

$\Rightarrow \frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=\frac{3b}{\alpha \beta \gamma }$
$\Rightarrow \frac{2}{\beta }+\frac{1}{\beta }=\frac{3b}{-c}(\because \alpha ,\beta ,\gamma inH.P\Rightarrow \frac{2}{\beta }=\frac{1}{\alpha }+\frac{1}{\gamma })$
$\Rightarrow \frac{1}{\beta }=-\frac{b}{c}$

$\Rightarrow \beta =\frac{-c}{b}$