If the roots of the equation x² + px + q = 0 are in the same ratio as those of the equation x² + lx + m = 0, then p²m is equal to

Let the roots of each equation be in the ratio k:1 .
Let k$\mathrm{\alpha }$ and o be the roots of x² + px + g = 0 .
Then $\mathrm{k\alpha }+\mathrm{\alpha }=-\mathrm{p} \mathrm{and} \mathrm{k\alpha }\cdot \mathrm{\alpha }=\mathrm{q}$
$\Rightarrow \mathrm{\alpha }=\frac{-\mathrm{p}}{\mathrm{k}+1}\text{ and }{\mathrm{\alpha }}^2=\frac{\mathrm{q}}{\mathrm{k}}$
$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\left(\frac{-\mathrm{p}}{\mathrm{k}+1}\right)}^2=\frac{\mathrm{q}}{\mathrm{k}}\\ \Rightarrow \frac{\mathrm{k}}{(\mathrm{k}+1{)}^2}=\frac{\mathrm{q}}{{\mathrm{p}}^2}......\left(\mathrm{i}\right)\end{array}$
Again, let $\mathrm{k\beta }$ and $\mathrm{\beta }$ be the roots of x² + lx+ m =0
Then, $\mathrm{k\beta }+\mathrm{\beta }=-\mathrm{I}\text{ and }\mathrm{k\beta }\cdot \mathrm{\beta }=\mathrm{m}$
$\Rightarrow \mathrm{\beta }=-\frac{1}{\mathrm{k}+1}\text{ and }{\mathrm{\beta }}^2=\frac{\mathrm{m}}{\mathrm{k}}$
$\begin{array}{l}\Rightarrow {\left(\frac{-1}{\mathrm{k}+1}\right)}^2=\frac{\mathrm{m}}{\mathrm{k}}\\ \Rightarrow \frac{\mathrm{k}}{(\mathrm{k}+1{)}^2}=\frac{\mathrm{m}}{{\mathrm{l}}^2}......\left(\mathrm{ii}\right)\end{array}$
Thus, from Eqs. (i) and (ii), we get
$\begin{array}{l}\frac{\mathrm{q}}{{\mathrm{p}}^2}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{\mathrm{m}}{{\mathrm{l}}^2}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{p}}^2\mathrm{m}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}={\mathrm{I}}^2\mathrm{q}\end{array}$