Mathematics Question from JEE Main 2025 · 28 January, Shift 2

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