Example 3 Let the vertices $A, B$ of $\triangle A O B$ in the complex plane correspond to the complex numbers $\alpha, \beta$, respectively, and satisfy: $\beta=(1+\mathrm{i}) \alpha,|\alpha-2|=$ $1, O$ is the origin, try to find the maximum and minimum values of the area $S$ of $\triangle O A B$.
Solve $\beta=(1+\mathrm{i}) \alpha=\sqrt{2}\left(\cos \frac{\pi}{4}+\mathrm{i} \sin \frac{\pi}{4}\right) \alpha,|\beta|=\sqrt{2}|\alpha|$, and by the geometric meaning of complex number multiplication, $\triangle A O B$ is an isosceles right triangle with $O B$ as the hypotenuse. $S_{\triangle \triangle O B}=\frac{1}{2} \cdot|\alpha| \times \sqrt{2}|\alpha| \times \sin 45^{\circ}=$ $\frac{1}{2}|\alpha|^{2}$, and the point corresponding to $\alpha$ is on a circle with center $(2,0)$ and radius 1, $1 \leqslant|\alpha| \leqslant 3$, i.e., the maximum and minimum values of $S_{\triangle A O B}$ are $\frac{9}{2}, \frac{1}{2}$.