Let \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) be three points in \(x y\)-plane, whose position vector are given by\(\sqrt{3…

Let \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) be three points in \(x y\)-plane, whose position vector 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 \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OB}}\) is \(\frac{9}{\sqrt{2}}\), then the sum of all the possible values of \(a\) is :

[JEE Main 2025, 28 Jan (Shift 2)]