Question

If roots of cubic equation $$a{x}^3+b{x}^2+cx+d$$ are in G.P., then:
$$\begin{array}{rl}& A.c^{3}a=b^{3}d\\ & B.a^{2}c=b^{2}d\\ & C.a{c}^2=b{d}^2\\ & D.\text{ NOTA}\end{array}$$

Answer 1

Hint: At first, take roots of the equation as $\alpha ,\alpha r,\alpha r^2$ with r as common ratio, then, use the relation between the roots and coefficient to get relation between a, b, c, d. Sum of roots is related to the ratio of –b and a of the quadratic equation whereas the product of roots is given by ratio of -d and a. Then we have another relation which gives the sum of product of two consecutive roots as ratio of c and a.

Complete step-by-step answer:
In the question, if the roots of cubic equation $$a{x}^3+b{x}^2+cx+d$$ are in G.P, then, we have to find a relation between a, b, c, d.
Now, as we know that, the roots are in G.P. So, let's suppose the roots are $$\alpha ,\alpha r,\alpha r^2$$ with a common ratio 'r'.
So, by relation between the roots and the coefficient of equation, we can say that,
Sum of the roots $$\Rightarrow -{\displaystyle \frac{b}{a}}$$
$$\Rightarrow \alpha +\alpha r+\alpha r^2=-{\displaystyle \frac{b}{a}}$$
Which can be written as,
$$\alpha (1+r+r^2)=-{\displaystyle \frac{b}{a}}$$
Product of the roots $$\Rightarrow {\displaystyle \frac{c}{a}}$$
$$\begin{array}{rl}& \Rightarrow \alpha \times \alpha r+\alpha \times \alpha r^2+\alpha r\times \alpha r^2={\displaystyle \frac{c}{a}}\\ & \Rightarrow {\alpha }^{2}r+{\alpha }^2{r}^2+{\alpha }^2{r}^3={\displaystyle \frac{c}{a}}\end{array}$$
Which can be written as,
$${\alpha }^{2}r(1+r+r^2)={\displaystyle \frac{c}{a}}$$
We can written it as,
$$\alpha r\times \alpha (1+r+r^2)={\displaystyle \frac{c}{a}}$$
Now, we can write $$\alpha (1+r+r^2)\text{ as -}{\displaystyle \frac{b}{a}}$$ as we know that $$\alpha (1+r+r^2)\text{ as -}{\displaystyle \frac{b}{a}}$$
$$\begin{array}{rl}& \alpha r\times -{\displaystyle \frac{b}{a}}={\displaystyle \frac{c}{a}}\\ & \Rightarrow \alpha r=-{\displaystyle \frac{c}{b}}\end{array}$$
The final relation between roots and coefficient of equation is,
$$\begin{array}{rl}& \alpha \times \alpha r\times \alpha r^2=-{\displaystyle \frac{d}{a}}\\ & \Rightarrow {\alpha }^3{r}^3=-{\displaystyle \frac{d}{a}}\end{array}$$
We know that, $$\alpha r=-{\displaystyle \frac{c}{b}}$$
So, we can write,
$${\alpha }^3{r}^3={\left(\alpha r\right)}^3\Rightarrow {(-{\displaystyle \frac{c}{b}})}^3$$
We found that $${\alpha }^3{r}^3\text{ is }-{\displaystyle \frac{d}{a}}$$ so, we can write,
$$\begin{array}{rl}& {(-{\displaystyle \frac{c}{b}})}^3=-{\displaystyle \frac{d}{a}}\\ & \Rightarrow -{\displaystyle \frac{c^3}{b^3}}=-{\displaystyle \frac{d}{a}}\end{array}$$
Now, on cross multiplication we get,
$$c^{3}a=b^{3}d$$
So, the correct answer is “Option A”.

Note: When it is said that, roots are given in G.P, then we can take the value of three roots as any three consecutive terms of any G.P such as $\alpha ,\alpha r,\alpha r^2$ by reducing the three variables into two. Also, while taking this, one should know here 'r' which is a common ratio cannot be equal to '0'. If 'r' is equal to '0' then, it will not satisfy the condition of being in G.P.