Example 2 Let $A$ be a fixed point $(b, 0)$, $P$ be a point on the circle $x^{2}+y^{2}=9$, and $M$ be a point on $AP$ such that $\frac{A M}{M P}=\frac{1}{2}$. When point $P$ moves on the circle, find the equation of the trajectory of point $M$.
Solution 1 As shown in Figure $27-1$, draw $M N / / P O$ intersecting the $x$-axis at $N$. It is evident that $|M N|$ is a fixed length, i.e., $|M N|=1$, and $N$ is a fixed point $(4,0)$.

By the plane geometry definition of a circle, the locus of points that are at a fixed distance from a fixed point is a circle, with the fixed point being the center and the fixed distance being the radius. Therefore, the equation of the required circle is $(x-4)^{2}+y^{2}=1$.

Solution 2 Let the coordinates of point $M$ be $(x, y)$, and let the parametric equations of the circle $x^{2}+y^{2}=9$ be $\left\{\begin{array}{l}x=3 \cos \theta, \\ y=3 \sin \theta\end{array}\right.$ ( $\theta$ is the parameter).

Thus, the coordinates of point $P$ can be set as $(3 \cos \theta, 3 \sin \theta)$. By the theorem in this example (or by the section formula for dividing a line segment in a given ratio), the parametric equations of the locus of point $M$ are $\left\{\begin{array}{l}x=4+\cos \theta, \\ y=\sin \theta .\end{array}(\theta\right.$ is the parameter).
From this, it is known that the locus of point $M$ on the line segment $A P$ is a circle with center at point $(4,0)$ and radius 1.