Question

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

• A. 2
• B. 1
• C. 3
• D. 4

Answer 1

Answer: (B)

Angle bisector is $x - y = 0$
$\Rightarrow \frac{\left|\beta -\left(1-\beta\right)\right|}{\sqrt{2}} = \frac{3}{\sqrt{2}} $
$ \Rightarrow \left|2\beta -1\right|=3 $
$ \Rightarrow \beta = 2$ or $ - 1 $