Let A, B, C be three points in x y -plane, whose position vector are given by √ 3 i + j, i +√ 3 j and a i +(1- a ) j respectively with respect to the…

Let $\displaystyle \mathrm{A}, \mathrm{B}, \mathrm{C}$ be three points in $\displaystyle x y$-plane, whose position vector are given by $\displaystyle \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\displaystyle \mathrm{a} \hat{i}+(1-\mathrm{a}) \hat{j}$ respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors $\displaystyle \overrightarrow{\mathrm{OA}}$ and $\displaystyle \overrightarrow{\mathrm{OB}}$ is $\displaystyle \frac{9}{\sqrt{2}}$, then the sum of all the possible values of a is :