Example 8 Let the complex number $z$ satisfy $\left|z-z_{1}\right|=\lambda\left|z-z_{2}\right|$, where $z_{1}, z_{2}$ are given distinct complex numbers, and $\lambda$ is a positive real constant. Try to discuss the locus of the point corresponding to the complex number $z$ in the complex plane.
Analyzing the use of $|z|^{2}=z \cdot \bar{z}$, we can transform $\left|z-z_{1}\right|^{2}=\lambda^{2}\left|z-z_{2}\right|^{2}$ to find the equation that $z$ satisfies.
Solution: We discuss in two cases:
(1) When $\lambda=1$, the curve of the equation is the perpendicular bisector of the line segment $Z_{1} Z_{2}$.
(2) When $\lambda \neq 1$, the equation is equivalent to $\left(z-z_{1}\right)\left(\bar{z}-\bar{z}_{1}\right)=\lambda^{2}\left(z-z_{2}\right)\left(\bar{z}-\bar{z}_{2}\right)$. Expanding and rearranging, we get $|z|^{2}-\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}} \bar{z}-\frac{\bar{z}_{1}-\lambda^{2} \bar{z}_{2}}{1-\lambda^{2}} z=\frac{\lambda^{2}\left|z_{2}\right|^{2}-\left|z_{1}\right|^{2}}{1-\lambda^{2}}$, which simplifies to
$$
\begin{array}{c}
|z|^{2}-\frac{z_{1}-\lambda^{2} z_{2}-\bar{z}}{1-\lambda^{2}}-\frac{\bar{z}_{1}-\lambda^{2} \bar{z}_{2}}{1-\lambda^{2}} z+\left|\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}\right|^{2} \\
=\frac{\lambda^{2}\left|z_{2}\right|^{2}-\left|z_{1}\right|^{2}}{1-\lambda^{2}}+\left|\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}\right|^{2}, \\
\text { Hence }\left|z-\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}\right|^{2}=\left|\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}\right|^{2}-\frac{\left|z_{1}\right|^{2}-\lambda^{2}\left|z_{2}\right|^{2}}{1-\lambda^{2}} \text {. }
\end{array}
$$

By the complex modulus inequality and the reverse Cauchy inequality, we know
$$
\left|\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}\right|^{2} \geqslant\left|\frac{\left|z_{1}\right|^{2}-\lambda^{2}\left|z_{2}\right|^{2}}{1-\lambda^{2}}\right| \geqslant \frac{\left|z_{1}\right|^{2}-\lambda^{2}\left|z_{2}\right|^{2}}{1-\lambda^{2}},
$$

Thus, the above equation is meaningful, and we have $\left|z-\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}\right|=\sqrt{\left|\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}\right|^{2}-\frac{\left|z_{1}\right|^{2}-\lambda^{2}\left|z_{2}\right|^{2}}{1-\lambda^{2}}}$. Therefore, when $\lambda \neq 1$, the trajectory of the point corresponding to $z$ in the complex plane is a circle with the center at the point corresponding to the complex number $z_{0}=\frac{z_{1}-\lambda^{2} z_{2}}{1-\lambda^{2}}$ and radius $r=\sqrt{\left|z_{0}\right|^{2}-\frac{\left|z_{1}\right|^{2}-\lambda^{2}\left|z_{2}\right|^{2}}{1-\lambda^{2}}}$.