Let A= [a_1; a_2] and B= [b_1; b_2] be two 2 × 1 matrices with real entries such that A=X B, where X =1/(√ 3) [1, -1; 1, k], and k ∈ R. If a_1^2+…

Let $\displaystyle A=\left[\begin{array}{l}a_1 \\ a_2\end{array}\right]$ and $\displaystyle B=\left[\begin{array}{l}b_1 \\ b_2\end{array}\right]$ be two $\displaystyle 2 \times 1$ matrices with real entries such that $\displaystyle A=X B$, where $\displaystyle \mathrm{X}=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & -1 \\ 1 & \mathrm{k}\end{array}\right]$, and $\displaystyle \mathrm{k} \in \mathrm{R}$. If $\displaystyle \mathrm{a}_1^2+\mathrm{a}_2^2=\frac{2}{3}\left(\mathrm{~b}_1^2+\mathrm{b}_2^2\right)$ and $\displaystyle \left(\mathrm{k}^2+1\right) \mathrm{b}_2^2 \neq-2 \mathrm{~b}_1 \mathrm{~b}_2$, then the value of k is \_\_\_\_.