Question:
Let $\sqrt{3} \hat{\mathrm{i}}+\hat{\mathrm{j}}, \hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}$ and $\beta \hat{\mathrm{i}}+(1-\beta) \hat{\mathrm{j}}$ respectively
be the position vectors of the points $A, B$ and $C$ with respect to the origin $\mathrm{O}$. If the distance of $\mathrm{C}$ from the bisector of the acute angle between OA
and $\mathrm{OB}$ is $\frac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is :-
Correct Option: , 2
Solution:
Angle bisector is $x-y=0$
$\Rightarrow \frac{|\beta-(1-\beta)|}{\sqrt{2}}=\frac{3}{\sqrt{2}}$
$\Rightarrow|2 \beta-1|=3$
$\Rightarrow \beta=2$ or $-1$