Example 9. As shown in Figure $9, \mathrm{AB}$ is the diameter of a fixed circle $\mathrm{O}$, and the moving chord $M N \perp A B$. Find the equation of the locus of the intersection point $P$ of the lines $A N$ and $M B$.
Solve: Take the line through B and A as the $x$-axis and the circle center $O$ as the origin to establish a Cartesian coordinate system. The moving point $\mathrm{P}$ is the intersection of line $\mathrm{AN}$ and $\mathrm{BM}$, with its coordinates set as $(x, y)$. Also, let the fixed circle radius be $\mathrm{R}$. The coordinates of each point are $\mathrm{A}(\mathrm{R}, 0)$, $B(-R, \hat{0})$, $M(R \cdot \cos \hat{0}, R \cdot \sin \theta)$, $N(R \cdot \cos (-0), R \sin (-\theta)$, where $\theta$ is a parameter. The equation of line $A N$ is: $y=\frac{-R \sin \theta}{R \cos \theta-R}(x-R)$. The equation of line $B M$ is: $y=\frac{R \sin \theta}{R \cos \theta+R}(x+R)$.
By solving (1) and (2) simultaneously, we get the parametric equations for $\mathrm{P}$.
Multiplying (1) by (2) and eliminating the parameter $\theta$, we obtain the ordinary equation as $x^{2}-y^{2}=R^{2}$.