Question

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

• A. $a{b}^3=c^{2}d$
• B. $a^{3}c=b{d}^3$
• C. $a{c}^3=b^{3}d$
• D. $a^{3}b=c{d}^2$

Answer 1

The correct option is C $a{c}^3=b^{3}d$

As roots are in G.P then let $\alpha ,\beta ,\gamma$ be the roots of $a{x}^3+b{x}^2+cx+d=0$
Such that $\alpha \gamma ={\beta }^2$
$S_3=\alpha \beta \gamma =-\frac{d}{a}\Rightarrow {\beta }^3=-\frac{d}{a}\Rightarrow \beta ={(-\frac{d}{a})}^{\frac{1}{3}}$
Substituting this in equation we get
$a{\left({(-\frac{d}{a})}^{\frac{1}{3}}\right)}^3+b{\left({(-\frac{d}{a})}^{\frac{1}{3}}\right)}^2+c{(-\frac{d}{a})}^{\frac{1}{3}}+d=0\Rightarrow -d+b{(-\frac{d}{a})}^{\frac{2}{3}}+c\left({(-\frac{d}{a})}^{\frac{1}{3}}\right)+d=0\Rightarrow a{c}^3=b^{3}d$