Question

The sum of the roots of the equation $a{x}^2+bx+c=0$is equal to the sum of the reciprocal of their squares, then $b{c}^2,c{a}^2,a{b}^2$will be in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is A

AP

Explanation for correct option

Step 1: The relation of roots

We know that if $\alpha ,\beta$ are the roots of a quadratic equation $r{x}^2+sx+t=0$ then

1. $\alpha +\beta =-\frac{s}{r}$
2. $\alpha \beta =\frac{t}{r}$

Let the roots of the equation $a{x}^2+bx+c$ be $p$and $q$

Therefore,

$p+q=\frac{-b}{a}$

$pq=\frac{c}{a}$

Given that,

$p+q=\frac{1}{p^2}+\frac{1}{q^2}$

Step 2: Find the required relation

$$\begin{gathered} p+q=\frac{p^2+q^2}{p^2{q}^2} \\ \Rightarrow p+q=\frac{{\left(p+q\right)}^2-2pq}{p^2{q}^2} \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{-b}{a}=\frac{{\left({\displaystyle \frac{-b}{a}}\right)}^2-{\displaystyle \frac{2c}{a}}}{{\left({\displaystyle \frac{c}{a}}\right)}^2} \\ \Rightarrow \frac{-b{c}^2}{a^3}=\frac{b^2}{a^2}-\frac{2c}{a} \\ \Rightarrow 2c{a}^2=a{b}^2+b{c}^2 \end{gathered}$$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{A}\right)$