Let $O$, $A$, $B$ be three non-collinear points on the coordinate plane, where the vector $\overrightarrow{OA}$ is perpendicular to $\overrightarrow{OB}$. If points $C$ and $D$ are on the line $AB$ satisfying $\overrightarrow{OC} = \frac{3}{5}\overrightarrow{OA} + \frac{2}{5}\overrightarrow{OB}$, $3\overline{AD} = 8\overline{BD}$, and $\overrightarrow{OC}$ is perpendicular to $\overrightarrow{OD}$, then $\frac{\overline{OB}}{\overline{OA}} = $\\
(Express as a fraction in lowest terms)