Example 9 (Zhou Jiansan). The line 1 passing through point $A_{2}(6,4)$ intersects the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ at points $\mathrm{M}, \mathrm{N}$. Find the maximum and minimum values of |AM|・|AN|.
Let the inclination angle be $\alpha$, then its parametric equation is
$$
\left\{\begin{array}{l}
x=6+t \sin \alpha, \\
y=4+t \cos \alpha .
\end{array}\right. \text { (t is a parameter) }
$$

Substituting into the ellipse equation, and simplifying, we get
$$
\begin{array}{l}
\left(3 \sin ^{2} \alpha+1\right) t^{2}+4(3 \cos \alpha+8 \sin \alpha)+84 \\
=0 .
\end{array}
$$

Thus,
$|A M| \cdot|A N|=\left|t_{1} t_{2}\right|$
$$
=\frac{84}{1+3 \sin ^{2} \alpha} \text {, }
$$
$\therefore(|\mathrm{AM}| \cdot|\mathrm{AN}|)_{\max }=84$
$(\alpha=0),(|\mathrm{AM}| \cdot|\mathrm{AN}|)_{\mathrm{m} 1 \mathrm{n}}=21$
$$
\left(\alpha=\frac{\pi}{2}\right) \text {. }
$$

Analysis: The point $(6,4)$ is outside the ellipse, so when $l$ intersects the ellipse, $\alpha$ should satisfy $\Delta=[4(3 \cos \alpha$
$$
\begin{array}{l}
+8 \sin \alpha)]^{2}-4\left(3 \sin ^{2} \alpha+1\right) \cdot 84 \geqslant 0(0 \leqslant \alpha \\
<\pi) .
\end{array}
$$
$<\pi$ ).
That is, $\frac{2}{\sqrt{29}} \leqslant \sin \alpha \leqslant \frac{2}{\sqrt{5}}$.

When finding the maximum and minimum values, this point was overlooked.