Example 21 Given a parabola $y^{2}=2 p x(p>0)$ with a chord $A B$ of length $l$, find the shortest distance from the midpoint $M$ of $A B$ to the $y$-axis, and the coordinates of point $M$ at this time.

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Given that $A$ and $B$ are on the parabola $y^{2}=2 p x$, we can set $A\left(\frac{a^{2}}{2 p}, a\right), B\left(\frac{b^{2}}{2 p}, b\right)$. Then the coordinates of the midpoint $M$ of $A B$ are $x_{M}=\frac{a^{2}+b^{2}}{4 p}, y_{M}=\frac{a+b}{2}$,
Thus, \((a+b)^{2}=4 y_{M}^{2}, (a-b)^{2}=2\left(a^{2}+b^{2}\right)-(a+b)^{2}=8 p x_{M}-4 y_{M}^{2}\).
Also, from the given condition \(|A B|=l\), we have
\[
\begin{aligned}
l^{2} & =\left(\frac{a^{2}-b^{2}}{2 p}\right)^{2}+(a-b)^{2}=(a-b)^{2} \cdot \frac{(a+b)^{2}+4 p^{2}}{4 p^{2}} \\
& =\left(8 p x_{M}-4 y_{M}^{2}\right) \cdot \frac{4 y_{M}^{2}+4 p^{2}}{4 p^{2}},
\end{aligned}
\]

which simplifies to \(4\left(2 p x_{M}-y_{M}^{2}\right)\left(y_{M}^{2}+p^{2}\right)-p^{2} l^{2}=0\).
(1) When \(l \geqslant 2 p\), from (1) we have
\[
\begin{aligned}
x_{M} & =\frac{1}{2 p}\left[\left(y_{M}^{2}+p^{2}\right)+\frac{p^{2} l^{2}}{4\left(y_{M}^{2}+p^{2}\right)}\right]-\frac{p}{2} \\
& \geqslant \frac{1}{2 p} \cdot 2 \sqrt{\frac{p^{2} l^{2}}{4}}-\frac{l}{2}=\frac{l-p}{2},
\end{aligned}
\]

The equality holds if and only if \(y_{M}^{2}+p^{2}=\frac{p^{2} l^{2}}{4\left(y_{M}^{2}+p^{2}\right)} \Leftrightarrow \sqrt{\frac{p(l-2 p)}{2}}\).
Therefore, the minimum distance from point \(M\) to the \(y\)-axis is \(\min x_{M}=\frac{l-p}{2}\), and at this point, the coordinates of \(M\) are \(\left(\frac{l-p}{2}, \sqrt{\left.\frac{p(l-2 p)}{2}\right)}\right.\).
(2) When \(0 < l < 2 p\), from (1) we have
\[
4\left(2 p x_{M}-y_{M}^{2}\right)\left(y_{M}^{2}+p^{2}\right)-p^{2} l^{2}=0 \Rightarrow 2 p x_{M}=\frac{y_{M}^{2}+p^{2}}{2}+\frac{p^{2} l^{2}}{8\left(y_{M}^{2}+p^{2}\right)}>0 \ (\text{since } 4 p^{4} \geqslant p^{2} l^{2}).
\]

It is clear that \(x_{M}\) is a monotonically increasing function of \(y_{M}^{2}\). When \(y_{M}=0\), \(x_{M}\) takes its minimum value \(\min x_{M}=\frac{l^{2}}{8 p}\), and at this point, the coordinates of \(M\) are \(\left(\frac{l^{2}}{8 p}, 0\right)\).
In summary, the shortest distance from \(M\) to the \(y\)-axis is
\[
\min x_{M}=\left\{\begin{array}{l}
\frac{l^{2}}{8 p}, \quad 0 < l < 2 p, \\
\frac{1}{2}(l-p), \quad l \geqslant 2 p.
\end{array}\right.
\]

When \(0 < l < 2 p\), the coordinates of \(M\) are \(\left(\frac{l^{2}}{8 p}, 0\right)\); when \(l \geqslant 2 p\), the coordinates of \(M\) are \(\left(\frac{l-p}{2}, \sqrt{\left.\frac{p(l-2 p)}{2}\right)}\right.\).