Question

If the root of the equation $a{x}^2+bx+c=0$ are in the ratio $m:n$ then

• A. $mn{a}^2=(m+n)c^2$
• B. $mn{b}^2=(m+n)ac$
• C. $mn{b}^2={(m+n)}^{2}ac$
• D. None of these

Answer 1

The correct option is C $mn{b}^2={(m+n)}^{2}ac$

Let's the roots of the equation $a{x}^2+bx+c=0$ are $\alpha$ and $\beta$

As the roots are in the ratio $m:n$, hence $\frac{\alpha }{\beta }=\frac{m}{n}$

Or we can say that $\alpha =km$ and $\beta =kn$, where $k$ is the proportionality constant.

Now the sum of the roots of the equation $\alpha +\beta =k(m+n)=\frac{-b}{a}$ ....$\left(1\right)$

and the product of the roots $\alpha .\beta =k^2\left(mn\right)=\frac{c}{a}$ ....$\left(2\right)$

By putting the value of $k$ from equation $\left(1\right)$ in to equation $\left(2\right)$, we get,

$\Rightarrow \frac{b^2\left(mn\right)}{(m+n{)}^2{a}^2}=\frac{c}{a}$

$\Rightarrow mn{b}^2=(m+n{)}^{2}ac$

Hence the correct answer is $C$