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

The given equation is $a{x}^2+bx+c=0$

Given ratio of the roots are $m:n$

Let the roots be $m\alpha$ and $n\alpha .$

then sum of the roots: $m\alpha +n\alpha =-\frac{b}{a}$

$\Rightarrow {(m+n)}^2{\alpha }^2=\frac{b^2}{a^2}$    …… ( 1 )

Product of the roots:  $\left(m\alpha \right)\left(n\alpha \right)=\frac{c}{a}$

$\Rightarrow mn{\alpha }^2=\frac{c}{a}$     …… ( 2 )

Dividing ( 1 ) by ( 2 ),

 we get $\frac{{(m+n)}^2{\alpha }^2}{mn{\alpha }^2}=\frac{\frac{b^2}{a^2}}{\frac{c}{a}}$

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