Question

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

The correct option is A

$c^{3}a=b^{3}d$

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

$b{\left(\frac{-d}{a}\right)}^{\frac{2}{3}}=c{\left(\frac{d}{a}\right)}^{\frac{1}{3}}$

Cubing both sides, we get

$b^3\frac{d^2}{a^2}=c^3\frac{d}{a}.$
$\Rightarrow b^{3}d=c^{3}a$