Example 1. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>$
0 ), find the locus of the midpoints of parallel chords with slope $k$.
Let the endpoints of the chord be \( P_{\mathrm{j}}\left(x_{\mathrm{j}}, y_{\mathrm{j}}\right) \) \((j=1,2)\), and the midpoint be \( P(x, y) \). Then we have
\[
\left\{\begin{array}{l}
\frac{x_{1}^{2}}{a^{2}}+\frac{y_{1}^{2}}{b^{2}}=1, \\
\frac{x_{2}^{2}}{a^{2}}+\frac{y_{2}^{2}}{b^{2}}=1, \\
x_{1}+x_{2}=2 x, \\
y_{1}+y_{2}=2 y, \\
\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=k .
\end{array}\right.
\]
Subtracting (2) from (1), we get
\[
\begin{array}{l}
b^{2}\left(x_{1}+x_{2}\right)\left(x_{1}-x_{2}\right)+a^{2}\left(y_{1}+y_{2}\right) \\
\cdot\left(y_{1}-y_{2}\right)=0,
\end{array}
\]

which simplifies to \( b^{2}\left(x_{1}+x_{2}\right)+a^{2}\left(y_{1}+y_{2}\right) \)
\[
\text { - } \frac{y_{1}-y_{2}}{x_{1}-x_{2}}=0 \text {. }
\]

Substituting (3), (4), and (5) into the above equation, we get
\[
2 b^{2} x+2 a^{2} y \cdot k=0 \text {, }
\]

which simplifies to \( b^{2} x+a^{2} k y=0 \).
Thus, the required trajectory is a line segment through the origin inside the ellipse.

Similarly, we can find the trajectory of the midpoints of parallel chords with slope \( k \) for the parabola \( y^{2}=2 p x \).