Question

The condition that the roots of $a{x}^3+b{x}^2+cx+d=0$ in H.P. is

• A. $2c^3+27a{d}^2=9bcd$
• B. $2b^3+27a^{2}d=9abc$
• C. $2b^3+27a{d}^2=9abc$
• D. $2c^3+27a^{2}d=9bcd$

Answer 1

Answer: (A)

$2c^3+27a{d}^2=9bcd$

Let $\alpha ,\beta ,\gamma$ be the roots of $a{x}^3+b{x}^2+cx+d=0$
$\alpha +\beta +\gamma =\frac{-b}{a}\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}\alpha \beta \gamma =\frac{-d}{a}$
As $\alpha ,\beta ,\gamma$ are in H.P
$\beta =\frac{3\alpha \beta \gamma }{\alpha \beta +\beta \gamma +\gamma \alpha }\Rightarrow \beta =\frac{-3d}{c}$

Substituting this in equation we get
$a{\left(\frac{-3d}{c}\right)}^3+b{\left(\frac{-3d}{c}\right)}^2+c\left(\frac{-3d}{c}\right)+d=0\Rightarrow 2c^3+27a{d}^2=9bcd$