If $\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$ and $2\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{b}$ . Then $\frac{\left|\overrightarrow{b}\right|}{\left|\overrightarrow{a}\right|}$ is

$\left(\overrightarrow{a}+\overrightarrow{b}\right).\overrightarrow{a}=0$

$a^2+\overrightarrow{b}.\overrightarrow{a}=0$

$\overrightarrow{b}.\overrightarrow{a}=-a^2$                 ……… (1)

$\left(2\overrightarrow{a}+\overrightarrow{b}\right).\overrightarrow{b}=0$

$2\left(\overrightarrow{a}.\overrightarrow{b}\right)+b^2=0$       ……… (2)

from (1) and (2)

$2\left(-a^2\right)+b^2=0$

$b^2=2a^2\Rightarrow \frac{b}{a}=\sqrt{2}$