Example 1. The moving circle $\mathrm{M}$ is externally tangent to the fixed circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4 \mathrm{x}=0$, and also tangent to the $\mathrm{y}$-axis. Find the equation of the locus of the center $\mathrm{M}$.
Analysis: As shown in Figure 1, the moving circle $\mathrm{M}(\mathrm{x}, \mathrm{y})$ is externally tangent to the fixed circle $\mathrm{O}_{1}(2,0)$ at $\mathrm{A}$, and is tangent to the $\mathrm{y}$-axis at B. Therefore, the condition for the motion of $\mathrm{M}$ must satisfy
$$
\left|\mathrm{O}_{1} \mathrm{M}\right|=|\mathrm{MB}|+2
$$
(2 is the radius of the fixed circle). Expressing this equation in terms of the coordinates $(\mathrm{x}, \mathrm{y})$ of $\mathrm{M}$, we have
$$
\sqrt{(x-2)^{2}+y^{2}}=x+2 \text {, }
$$

Simplifying this to $\mathrm{y}^{2}=8 \mathrm{x}$, this is the equation of the trajectory of $\mathrm{M}$.