10. (20 points) Draw any line through the fixed point $M(m, 0)$, intersecting the parabola $y^{2}=16 x$ at points $P$ and $Q$. If $\frac{1}{|P M|^{2}}+\frac{1}{|Q M|^{2}}$ is a constant, find all possible values of the real number $m$.


10. Let the equation of the line passing through point $M(m, 0)$ be
$$
x=m+\lambda y \text {. }
$$

As shown in Figure 3.
From $|P M| \neq 0,|Q M| \neq 0 \Rightarrow m \neq 0$.
Let $P\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right)$. Then $y_{1} 、 y_{2}$ are the two roots of the equation $y^{2}=16(m+\lambda y)$.
Thus, $y_{1}+y_{2}=16 \lambda, y_{1} y_{2}=-16 m$
$$
\begin{aligned}
\Rightarrow & \frac{1}{y_{1}^{2}}+\frac{1}{y_{2}^{2}}=\frac{\left(y_{1}+y_{2}\right)^{2}-2 y_{1} y_{2}}{y_{1}^{2} y_{2}^{2}} \\
& =\frac{16^{2} \lambda^{2}+32 m}{16^{2} m^{2}}=\frac{16 \lambda^{2}+2 m}{16 m^{2}}, \\
& |P M|^{2}=\left(x_{1}-m\right)^{2}+y_{1}^{2} \\
& =\lambda^{2} y_{1}^{2}+y_{1}^{2}=\left(1+\lambda^{2}\right) y_{1}^{2}, \\
& |Q M|^{2}=\left(1+\lambda^{2}\right) y_{2}^{2} .
\end{aligned}
$$

Then $\frac{1}{|P M|^{2}}+\frac{1}{|Q M|^{2}}$
$=\frac{1}{1+\lambda^{2}}\left(\frac{1}{y_{1}^{2}}+\frac{1}{y_{2}^{2}}\right)=\frac{16 \lambda^{2}+2 m}{\lambda^{2}+1} \cdot \frac{1}{16 m^{2}}$
$=\left(16+\frac{2 m-16}{\lambda^{2}+1}\right) \frac{1}{16 m^{2}}$.
From the fact that equation (1) is a constant, we know that $\frac{2 m-16}{\lambda^{2}+1}$ does not change with $\lambda$, i.e.,
$$
2 m-16=0 \Rightarrow m=8 ;
$$

Conversely, if $m=8$, then equation (1) is a constant $\frac{1}{64}$.
Therefore, $m=8$.