Let √ 3 i + j, i +√ 3 j and β i +(1-β) j respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance…

Let $\displaystyle \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\displaystyle \beta \hat{i}+(1-\beta) \hat{j}$ respectively be the position vectors of the points A, B and C with respect to the origin $\displaystyle O$. If the distance of $\displaystyle C$ from the bisector of the acute angle between $\displaystyle O A$ and $\displaystyle O B$ is $\displaystyle \frac{3}{\sqrt{2}}$, then the sum of all possible values of $\displaystyle \beta$ is :