Question

If the sum of the roots of the quadratic equation $a{x}^2+bx+c=0$ is equal to the sum of the squares of their reciprocals then $\frac{b^2}{ac}+\frac{bc}{a^2}$ is equal to

• A. $2$
• B. $-2$
• C. $1$
• D. $-1$

Answer 1

The correct option is A $2$

Let $\alpha$ and $\beta$ are the roots of given equation.

We have $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$,

Now according to the given condition in question, we have

$\alpha +\beta =\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$.

On further solving it, we have

$\alpha +\beta =\frac{({\alpha }^2+{\beta }^2)}{(\alpha \beta {)}^2}$

$\alpha +\beta =\frac{(\alpha +\beta {)}^2-2\alpha \beta }{(\alpha \beta {)}^2}$

Substituting the value of $\alpha +\beta$ and $\alpha \beta$

$-\frac{b}{a}=\frac{(\frac{b^2}{a^2}-2\frac{c}{a})}{\frac{c^2}{a^2}}$

$\Rightarrow -\frac{b}{a}=\frac{(b^2-2ac)}{c^2}$

$\Rightarrow 2ac=b^2+\frac{b{c}^2}{a}$

$\Rightarrow 2=\frac{b^2}{ac}+\frac{bc}{a^2}$

Hence, option A is correct.