9. (20 points) Given a point $A(0, a)(a>1)$ on the $y$-axis, for a moving point $M(x, y)$ on the curve $y=$ $\left|\frac{1}{2} x^{2}-1\right|$, try to find the minimum distance $|A M|$ between points $A$ and $M$ (expressed in terms of $a$).
9 As shown in the figure, the coordinates of the points on the curve can be easily obtained as:
$$
E(-\sqrt{2}, 0), F(\sqrt{2}, 0), D(0,1) \text {, when }
$$
$x^{2}1$, then $2 a>2, \sqrt{2 a}>\sqrt{2}$, thus, when $x=\sqrt{2 a}$, $|A M|$ reaches its minimum value $\sqrt{2 a+1}$; then compare $A D$ with $A M$, let
$$
f(a)=|A D|^{2}-|A M|^{2}=(a-1)^{2}-(2 a+1)=a(a-4) \text {, }
$$

then when $14$, $f(a)>0$, then the minimum value $|A M|=\sqrt{2 a+1}$.
Therefore, $|A M|_{\text {min }}=\left\{\begin{array}{cc}a-1, & 14 .\end{array}\right.$