Find the locus equation of points whose absolute value of the difference of distances to the fixed points $F_{1}(-c, 0), F_{2}(c, 0)$ is $2a$ $(c>a>0)$.
Let $M(x, y)$ be any point on the trajectory, we can obtain the system of equations
$$
=\left\{\begin{array}{l}
(x+c)^{2}+y^{2}=r_{1}^{2} \\
(x-c)^{2}+y^{2}=r_{2}^{2}, \\
\left|r_{1}-r_{2}\right|=2 a .
\end{array}\right.
$$

Thus, $\left\{\begin{array}{l}r_{1}+r_{2}=\frac{2 c x}{ \pm a} \\ r_{1}-r_{2}= \pm 2 a\end{array}\right.$
Therefore, $\quad r_{1}= \pm\left(\frac{c x}{a}+a\right)$.
Substituting the value of $r_{1}$ into the first equation of the system, and letting $c^{2}-a^{2}=b^{2}$, we get
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}-1
$$