Question

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

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

Answer 1

Answer: (A)

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