Question

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

• A. $c^2\left(mn\right)=ab(m+n{)}^2$
• B. $b^2(m+n)=mn$
• C. $mn{b}^2=ac(m+n{)}^2$
• D. $m+n=b^{2}mn$

Answer 1

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

Given: Roots of the equation $a{x}^2+bx+c=0$ are in the ratio $m:n$
Let roots be $\alpha ,\beta$
$\alpha +\beta =\frac{-b}{a},\alpha .\beta =\frac{c}{a},\frac{\alpha }{\beta }=\frac{m}{n}$

Using componendo and dividendo in $\frac{\alpha }{\beta }=\frac{m}{n}$
$\Rightarrow \frac{\alpha +\beta }{\alpha -\beta }=\frac{m+n}{m-n}\Rightarrow \frac{(\alpha +\beta {)}^2}{(\alpha +\beta {)}^2-4\alpha \beta }=\frac{(m+n{)}^2}{(m+n{)}^2-4mn}\Rightarrow \frac{4\alpha \beta }{(\alpha +\beta {)}^2}=\frac{4mn}{(m+n{)}^2}\Rightarrow \frac{c/a}{b^2/a^2}=\frac{mn}{(m+n{)}^2}\Rightarrow ac(m+n{)}^2=mn{b}^2$