Proposition 2 Find the equation of the locus of the midpoints of parallel chords with slope $k$ of the conic section $f(x, y)=0$.

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From the proof of Proposition 1, equation (3) shows that when the line varies, the midpoint \( P_{\mathrm{E}}\left(x_{0}^{\prime}\right., y_{0} \) of the chord it intercepts also varies. Therefore, equation (3) becomes the relationship that the coordinates of the midpoints of the chords should satisfy:
$$
\begin{array}{c}
(2 A x+B y+D) \cos \theta \\
+(B x+2 C y+E) \sin \theta=0 .
\end{array}
$$

Since the slope \( k \) of the parallel chords exists, i.e., \( \cos \theta \neq 0 \), dividing equation (3') by \( \cos \theta \) yields
$$
2 A x+B y+D+(B x+2 C y+E) k=0 .
$$

Expanding and rearranging gives \((2 A+B k) x+(B+2 C k) y\)
$$
+D+E k=0 .
$$
Equation (8) is the equation of the locus of the midpoints of the parallel chords with slope \( k \).