Question

If the ratio of the roots of $a{x}^2+2bx+c=0$ is same as the roots of $p{x}^2+2qx+r=0$ then:

• A. $b/ac=q/pr$
• B. $b^2/ac=q^2/pr$
• C. $2b/ac=q^2/pr$
• D. none of these

Answer 1

The correct option is B $b^2/ac=q^2/pr$

Let $\alpha ,\beta$ be roots of $a{x}^3+bx+c=0$

$\gamma ,\delta$ be roots of $p{x}^2+2qx+r=0$

$\frac{\alpha }{\beta }=\frac{\gamma }{\delta }$ …………$\left(1\right)$

and $\frac{\beta }{\alpha }=\frac{\delta }{\gamma }$ ………….$\left(2\right)$

$\left(1\right)+\left(2\right)$

$\Rightarrow \frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{\gamma }{\delta }+\frac{\delta }{\gamma }$

$=\frac{2^2+{\beta }^2}{\alpha \beta }+2=\frac{{\gamma }^2+{\delta }^2}{\gamma \delta }+2$

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

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

$\Rightarrow \frac{\frac{4b^2}{a^2}}{\frac{c}{a}}=\frac{\frac{4q^2}{p^2}}{\frac{r}{p}}$

$\Rightarrow \frac{b^2}{ac}=\frac{q^2}{p^r}$.