Example 4. Find the locus of the midpoint of the chord passing through the imaginary vertex $B(0,-b)$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$,
Let $M (x, y)$ be the midpoint of the chord that meets the condition, then we have
$$
k_{B \boldsymbol{\mu}}=\frac{y+b}{x}, k_{\circ}=\frac{y}{x} .(x \neq 0)
$$

By $k_{B M} \cdot k_{o_{\mathrm{M}}}=\frac{b^{2}}{a^{2}}$, i.e., $\frac{y+b}{x} \cdot \frac{y}{x}$
$$
=\frac{b^{2}}{a^{2}},
$$

we get $\quad b^{2} x^{2}-a^{2} y^{2}-a^{2} b y=0$.
Therefore,
$$
\frac{\left(y+\frac{b}{2}\right)^{2}}{\frac{b^{2}}{4}}-\frac{x^{2}}{\frac{a^{2}}{4}}=1 .
$$

This is a hyperbola with the center at $\left(0,-\frac{b}{2}\right)$ and foci on the $y$-axis. (Note: The problem should also discuss the singularity of the trajectory, which is omitted here)