Example 1. Given the line $\mathrm{y}=\mathrm{kx}+\mathrm{m}$ and the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ are tangent. Try to determine the range of $k$.
Solve $\left\{\begin{array}{l}y=k x+m, \\ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\end{array}\right.$
Substitute (1) into (2), and rearrange to get
$$
\begin{array}{l} 
\quad\left(a^{2} k^{2}-b^{2}\right) x^{2}+2 a^{2} k m x+a^{2}\left(m^{2}\right. \\
\left.+b^{2}\right)=0 .
\end{array}
$$
Since the line is tangent to the hyperbola,
Therefore, the discriminant $\Delta=0$ of equation (3), i.e., $\left(2 a^{2} k m\right)^{2}-4\left(a^{2} k^{2}-b^{2}\right) \cdot a^{2}\left(m^{2}\right.$ $+b^{2}$ ) $=0$.

Rearrange to get $k^{2} a^{2}-b^{2}=m^{2}$. Then $k^{2}=\frac{b^{2}}{a^{2}}+\frac{m^{2}}{a^{2}}$.
1. When $m=0$, $k^{2}=\frac{b^{2}}{a^{2}}$, then $k= \pm \frac{b}{a}$, at this time the line $y=k x+m$ is $y= \pm \frac{b}{a} x$,
which are the asymptotes of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Clearly, the line does not touch the hyperbola (in the general sense).
2. When $m \neq 0$, $k^{2}>\frac{b^{2}}{a^{2}}$, which means $k>$ $\frac{b}{a}$ or $k<-\frac{b}{a}$ (or $m \neq 0$ );
(2) When $|k|<\frac{b}{a}$ (i.e., $-\frac{b}{a}<k<\frac{b}{a}$), the line $y=k x+m$ is not a tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$;
(3) Incidentally, let's add a bit of supplement to a problem in the textbook. Problem 21 on page 171 is:
"Prove: The condition for the line $y=k x+m$ to be tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is
$$
k^{2} a^{2}-b^{2}=m^{2} \text{. }
$$

From the solution of Example 1, it is easy to see that the condition $k^{2} a^{2}-b^{2}=m^{2}$ is not a sufficient and necessary condition for the line $y=k x+m$ to be tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, but only a necessary condition. This means that the condition $k^{2} a^{2}-b^{2}=m^{2}$ alone cannot ensure that the line is tangent to the hyperbola, so this problem is not quite appropriate. It is suggested to change it to:
"Given: The line $y=k x+m$ is tangent to the hyperbola
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Prove $k^{2} a^{2}-b^{2}=m^{2}$."