Example 4. The locus of the midpoints of parallel chords of the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}(a>0$, b>0) is.
Given the equation (8), with $A=b^{2}, C=a^{2}$, $B=D=E=0, F=-a^{2} b^{2}$, the required trajectory equation is $2 b^{2} x+2 a^{2} k y=0$.

Therefore, when $k \neq 0$, it is $y=-\frac{b^{2}}{a^{2} k} x$; when $k=0$, it is $x=0$. They are all straight line equations passing through the center of the ellipse (i.e., the origin). The trajectory of the midpoint of the chord is the line segment intercepted by the ellipse.
3. Find the trajectory equation of the midpoint of a moving chord with a fixed length
Proposition 3 Find the trajectory equation of the midpoint of a moving chord with a fixed length $m$ for the conic section $f(x, y)=0$.

Solution From the solution of Proposition 2, we know that the coordinates of the midpoints of the chords should satisfy the relation $(2 A x+B y+D) \cos \theta + (B x+2 C y+E) \sin \theta=0$.
Also, from the chord length formula $\left(L_{A B}\right)$, we have $m=\sqrt{\left(t_{1}+t_{2}\right)^{2}-4 t_{1} t_{2}}$, and $t_{1}+t_{2}=\mathrm{C}$,
$$
\therefore m^{2}=-4 t_{1} t_{2} \text{. }
$$

From equation (2) and letting the midpoint coordinates be $(x, y)$, we have
$$
t_{1} \cdot t_{2}=\frac{A x^{2}+B x y+C y^{2}+D x+E y+F}{A \cos ^{2} \theta+B \cos \theta \sin \theta+C \sin ^{2} \theta} .
$$

Therefore, the coordinates of the midpoint of a chord with a fixed length $m$ should satisfy the system of equations
$$
\left\{\begin{array}{l}
(2 A x+B y+D) \cos \theta+(B x+2 C y+E) \sin \theta=0, \\
m^{2}=-4 \times \frac{A x^{2}+B x y+C y^{2}+D x+E y+F}{A \cos ^{2} \theta+B \cos \theta \sin \theta+C \sin ^{2} \theta} .
\end{array}\right.
$$

By eliminating $\sin \theta, \cos \theta$ in (10), we obtain the trajectory equation of the point with a moving chord of fixed length $m$.

According to the system of equations (10), we can find that: for a line segment $P Q$ with a fixed length $n$ $(n \geqslant 1)$, whose endpoints move on the parabola $y=x^{2}$, the trajectory equation of the midpoint of the line segment $P Q$ is
$$
16 x^{4}-16 x^{2} y+4 x^{2}-4 y+n^{2}=0 \text{. }
$$
(Author's unit: Department of Mathematics, Huazhong Normal University)