设矩阵 $A =\left(\begin{array}{cc}1 & 2 \\ -2 & -a\end{array}\right), B =\left(\begin{array}{ll}1 & 0 \\ 1 & a\end{array}\right)$, 若 $f(x, y)=|x A +y B |$ 是正定二次型, 则 $a$ 的取值范围是
A. $(0,2-\sqrt{3})$
B. $(2-\sqrt{3}, 2+\sqrt{3})$
C. $(2+\sqrt{3}, 4)$
D. $(0,4)$