# JEE-Advanced 2015 (20.C/40)

Geometry Level 3

In $$\mathbb R^2$$, let $$\sqrt3 \hat i +\hat j \ \ , \ \hat i +\sqrt3 \hat j$$ and $$\beta \hat i +(1-\beta) \hat j$$ be the position vectors of $$X,Y$$ and $$Z$$ with respect to the origin $$O$$, respectively. If the distance of $$Z$$ from the bisector of the acute angle of $$\vec{OX}$$ with $$\vec{OY}$$ is $$\frac{3}{\sqrt2}$$, then find the possible value(s) of $$|\beta|$$. $\begin{array}{} (1) \, 1 \quad \quad \quad \quad \quad \quad \quad \quad & (2) \, 2 \\ (3) \, 3 & (4) \, 5 \end{array}$ Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $$(1),(2)$$, then submit 12 as the correct answer, if your answer is $$(2),(3),(4)$$, then submit 234 as the correct answer.

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