If the ratio of the roots of the equation $x^2+px+q=0$  be same as that of the roots of the equation $x^2+lx+m=0$  then

The given equations are $x^2+px+q=0$ ……. ( 1 )

And $x^2+lx+m=0$  …… ( 2 )

Let the roots of the two equations be

$\alpha ,\beta$  and $\gamma ,\delta$  respectively,

From Eq.( 1 ):

Some of the roots  $\alpha +\beta =-p,$

Product of the roots $\alpha \beta =q$

From Eq.( 2 ):

Some of the roots  $\gamma +\delta =-l,$

Product of the roots $\gamma \delta =m$

By the given condition, 

 $\frac{\alpha }{\beta }=\frac{\gamma }{\delta }\Rightarrow =\frac{\alpha +\beta }{\alpha -\beta }=\frac{\gamma +\delta }{\gamma -\delta }$

Squaring on both sides

$\Rightarrow \frac{{\left(\alpha +\beta \right)}^2}{{\left(\alpha -\beta \right)}^2}=\frac{{\left(\gamma +\delta \right)}^2}{{\left(\gamma -\delta \right)}^2}$

$\Rightarrow \frac{{\left(\alpha +\beta \right)}^2}{{\left(\alpha +\beta \right)}^2-4\alpha \beta }=\frac{{\left(\gamma +\delta \right)}^2}{{\left(\gamma +\delta \right)}^2-4\gamma \delta }$

$\Rightarrow \frac{p^2}{p^2-4q}=\frac{l^2}{l^2-4m}$

$\Rightarrow p^2{l}^2-4p^{2}m=l^2{p}^2-4l^{2}q$

$\Rightarrow p^{2}m=l^{2}q.$