Tangents

Let the angle between the lines $y=x$ and $y=2x$ be $\theta$ , then $\theta = \tan^{-1}2-\frac \pi4$ and $\tan \theta = \dfrac {2-1}{1+2} = \dfrac 13$ . Then we have:

$\begin{aligned} \frac {2\tan \frac \theta 2}{1- \tan^2 \frac \theta 2} & = \tan \theta = \frac 13 \\ \tan^2 \frac \theta 2 + 6\tan \frac \theta 2 - 1 & = 0 \\ \implies \tan \frac \theta 2 & = \frac {-6 \pm \sqrt{36+4}}2 \\ & = \sqrt{10}-3 & \small \color{#3D99F6} \text{Note that } \tan \frac \theta 2 > 0 \end{aligned}$

We note that:

$\begin{aligned} \tan \frac \theta 2 & = \frac 3{|OA|} \\ \implies |OA| & = \frac 3{\tan \frac \theta 2} \\ & = \frac 3{\sqrt{10}-3} \\ \\ & = \frac {3(\sqrt{10}+3)}{(\sqrt{10}-3)(\sqrt{10}+3)} \\ & = 9 + 3\sqrt{10} \end{aligned}$

$\implies a+b = 9+3 = \boxed{12}$