$\text{ If the line }x+2y+4=0\text{ is a common tangent to the parabolas }{y}^2=4ax\text{ and }{x}^2=4by,\text{ then }|b-a|\text{ is }$

$\text{ Tangent to parabola }{y}^2=4ax\text{ having slope }m\text{ is }y=mx+\frac{a}{m}\text{ . }$

$\text{ It also touches parabola }{x}^2=4by\text{ . }$

$\begin{array}{l}\therefore x^2=4b\left(mx+\frac{a}{m}\right)\text{ has equal roots. }\\ \therefore 16b^2{m}^2+16\frac{ab}{m}=0\\ \therefore b{m}^3+a=0\end{array}$

$\begin{array}{l}\therefore m=-{\left(\frac{a}{b}\right)}^{1/3}\\ \therefore \text{ The equation of common tangent is }\\ y=-{\left(\frac{a}{b}\right)}^{1/3}x-a{\left(\frac{b}{a}\right)}^{1/3}\end{array}$

$\begin{array}{l}\text{ or }{a}^{\frac{1}{3}}x+b^{\frac{1}{3}}y+a^{\frac{2}{3}}{b}^{\frac{2}{3}}=0\\ \text{ Comparing with }x+2y+4=0\\ a=1,b=8\\ \Rightarrow |b-a|=7\end{array}$