If the ratio of the roots of  $a{x}^2+2bx+c=0$ is same as the ratio of the roots of  $p{x}^2+2qx+r=0$ then  $b^2:q^2$ is

Given  $a{x}^2+2bx+c=0$

$p{x}^2+2qx+r=0$

The ratio of roots same
$\frac{a}{p}=\frac{2b}{2q}=\frac{c}{r}$

$\frac{a}{p}=\frac{b}{q}=\frac{c}{r}$
$\text{Let\hspace{0.17em}\hspace{0.17em}}\frac{a^2}{p^2}=\frac{b^2}{q^2}=\frac{c^2}{r^2}=k$

$a^2=p^{2}k,c^2=r^{2}k$

$a^2{c}^2=p^2{r}^2{k}^2\Rightarrow k^2=\frac{a^2{c}^2}{p^2{r}^2}$

$\frac{b^2}{q^2}=\frac{ac}{pr}$