Question

If the roots of $a{x}^2+bx+c=0$ 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 $mn{b}^2={(m+n)}^{2}ac$

$\frac{\alpha }{\beta }=\frac{m}{n}$
or by ratio proportion theory,
$\frac{\alpha }{m}=\frac{\beta }{n}=\frac{\alpha +\beta }{m+n}=\sqrt{\left(\frac{\alpha \beta }{mn}\right)}$
$\therefore mn{(\alpha +\beta )}^2=\alpha \beta {(m+n)}^2$
or $mn{(-\frac{b}{a})}^2={(m+n)}^2\frac{c}{a}$
or $mn{b}^2=ac{(m+n)}^2$