Example. Find the equation of the locus of the midpoints of the parallel chords with slope $m$ of the parabola $y^{\prime \prime}=2 p x$.

保留源文本的换行和格式，翻译结果如下：

Example. Find the equation of the locus of the midpoints of the parallel chords with slope $m$ of the parabola $y^{\prime \prime}=2 p x$.
Solution: According to the proposition, the trajectory is a ray parallel to the $y$-axis. Taking a chord with slope $m$, one of which is $y = mx$, and substituting it into $y^{2} = 2px$, we find the intersection points $O(0,0)$ and $P\left(\frac{2p}{m^{2}}, \frac{2p}{m}\right)$. The midpoint of $OP$ is $\left(\frac{p}{m}\right)$, so the required equation is $y = \frac{p}{m} . \quad\left(x \geqslant \frac{p}{2m}\right)$