Example 2. Let point $P(x, y)$ be such that the sum of its distances to two foci $F(-c, 0)$ and $\pi(c, 0)$ is a constant $2a (a > 0)$. Find the parametric equations of the locus of point $P$.
Solution: Connect $P F_{1}, P F_{2}$, and draw $P S \perp F_{1} F_{2}$ at $S$ (Figure 2), forming the right triangle $\triangle P F_{2} S$. Let $\angle P F_{2} S = \theta$ be the parameter, then $\left\{F_{z}\right.$ !
$=c-x$, and $\left|F_{2} S\right|$
$=\left|P F_{2}\right| \cos \theta$,
$\therefore c-x=(a-e x) \cos \theta$,
$x=\frac{a \cos \theta-e}{e \cos \theta-1}=\frac{a(a \cos \theta-c}{c \cos \theta-a}$
$=\frac{a(c-a \cos \theta)}{a-c \cos \theta}$.

Similarly, we get $y=\frac{b^{2} \sin \theta}{a-c \cos \theta}$.