Question

Find the relation if the relation if the roots of $a{x}^2+bx+c=0$ are in the ratio $m:n$.

Answer 1

The roots are in the ratio $m:n$ then roots are my and ny.

Sum of roots$=my+ny$

$y(m+n)=\frac{-b}{a}$

Product of roots $=\left(my\right)\left(ny\right)$

$\left(mn\right)y^2=\frac{c}{a}$

According to problem

$\left[y\right(m+n\left){]}^2/\right(mn)y^2=\frac{(-b/a{)}^2}{c/a}$

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

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

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