Question

If the roots of the cubic equation $a{x}^3+b{x}^2+cx+d=0$ are in GP, then

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

Answer 1

Answer: (A)

Let $\frac{A}{R},A,AR$ be the roots of the equation
$a{x}^3+b{x}^2+cx+d=0$, then Product of roots $<br/><br/>A^3=-\frac{d}{a}$
$\Rightarrow =A=-{\left(\frac{d}{a}\right)}^{1/3}$
Since, $A$ is a root of the equation.
$\therefore a{A}^3+b{A}^2+cA+d=0$
$\Rightarrow a\left(-\frac{d}{a}\right)+b{\left(-\frac{d}{a}\right)}^{2/3}+c{\left(-\frac{d}{a}\right)}^{1/3}+d=0$
$\Rightarrow b{\left(\frac{d}{a}\right)}^{2/3}=c{\left(\frac{d}{a}\right)}^{1/3}$
$\Rightarrow b^3\cdot \frac{d^2}{a^2}=c^3\cdot \frac{d}{a}$
$\Rightarrow b^{3}d=c^{3}a$