If the solve the problem

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 :-

  1. 2

  2. 1

  3. 3

  4. 4

Correct Option: , 2


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 $-1$


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