Equation of line touching both parabolas ${\mathrm{y}}^2=4\mathrm{x}$ and ${\mathrm{x}}^2=-32\mathrm{y}$ is

$\begin{array}{l}{\mathrm{y}}^2=4\mathrm{x}\Rightarrow {\mathrm{x}}^2=-32\mathrm{y}\Rightarrow {\mathrm{y}}^2=4\mathrm{ax}\Rightarrow {\mathrm{x}}^2=4\mathrm{by}\\ \Rightarrow \mathrm{a}=1\Rightarrow \mathrm{b}=-8\end{array}$
eqn of common tgt is ${\mathrm{xa}}^{1/3}+{\mathrm{yb}}^{1/3}+{\mathrm{a}}^{2/3}{\mathrm{b}}^{2/3}=0$
$\begin{array}{l}\Rightarrow \mathrm{x}(1{)}^{1/3}+\mathrm{y}(-8{)}^{1/3}+\left(1{)}^{2/3}\right(-8{)}^{2/3}=0\\ \Rightarrow \mathrm{x}-2\mathrm{y}+4=0\end{array}$