5. As shown in Figure 1, given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $(a>0, b>0), A A^{\prime}$ is a chord passing through the right focus $F$ and perpendicular to the real axis $x$, $M$ is the right vertex of the hyperbola, and $\angle A M A^{\prime}=\alpha$. Then, ( ).
(A) $\alpha$ could be $90^{\circ}$
(B) $\alpha$ could be $120^{\circ}$
(C) $90^{\circ}<\alpha<120^{\circ}$
(D) $120^{\circ}<\alpha<180^{\circ}$
5.D.

Point $F\left(\sqrt{a^{2}+b^{2}}, 0\right)$, then $A\left(\sqrt{a^{2}+b^{2}}, \frac{b^{2}}{a}\right)$.
Thus, $\tan \frac{a}{2}=\frac{A F}{M F}=\frac{\frac{b^{2}}{a}}{\sqrt{a^{2}+b^{2}}-a}=\frac{\sqrt{a^{2}+b^{2}}+a}{a}$ $>2>\sqrt{3}$.

Therefore, $60^{\circ}<\frac{\alpha}{2}<90^{\circ}$, which means $120^{\circ}<\alpha<180^{\circ}$.