Matrices and Determinants: JEE Main 2021 Mathematics question

Let $A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]$ and $B = \left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr } } \right]$ be two 2 $\times$ 1 matrices with real entries such that A = XB, where $X = {1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & { - 1} \cr 1 & k \cr } } \right]$, and k$\in$R. If $a_1^2$ + $a_2^2$ = ${2 \over 3}$(b$_1^2$ + b$_2^2$) and (k² + 1) b$_2^2$ $\ne$ $-$2b₁b₂, then the value of k is __________.