Let 3 i ^ + j ^ , i ^ + 3 j ^ and β i ^ + ( 1 − β ) 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 o

Question

Let 3 ​ i ^ + j ^ ​ , i ^ + 3 ​ j ^ ​ and β i ^ + ( 1 − β ) 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 2 ​ 3 ​ , then the sum of all possible values of β is:

TestHub · Accepted Answer

Since, the angle bisector of acute angle between $O A$ and $OB$ would be $y = x$
Since, the distance of $C$ from bisector $= \frac{3}{2}$ $\Rightarrow \frac{β - ( 1 - β )}{2} = \frac{3}{2} = 2 β = \pm 3 + 1$ $β = 2$ or $β = - 1$ Hence, the sum of all possible value of $β = 2 + (- 1) = 1$

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