10. (20 points) As shown in Figure 3, the ellipses $C_{1}$ and $C_{2}$, with centers at the origin and foci on the $x$-axis and $y$-axis respectively, both pass through the point $M(0, -\sqrt{2})$, and the eccentricities of ellipses $C_{1}$ and $C_{2}$ are both $\frac{\sqrt{2}}{2}$. Two lines with slopes $k$ and $k'$ are drawn through point $M$, intersecting ellipses $C_{1}$ and $C_{2}$ at points $P$ and $Q$ respectively. When $k' = 4k$, does the line $PQ$ pass through a fixed point? If it does, find the coordinates of the fixed point; if not, explain the reason.
10. Notice that the standard equations of ellipses $C_{1}$ and $C_{2}$ are
$$
\frac{x^{2}}{4}+\frac{y^{2}}{2}=1, \frac{y^{2}}{2}+x^{2}=1 \text {; }
$$

The line $l_{M P}: y=k x-\sqrt{2}$.
From $\left\{\begin{array}{l}\frac{x^{2}}{4}+\frac{y^{2}}{2}=1, \\ y=k x-\sqrt{2}\end{array}\right.$
$\Rightarrow\left(2 k^{2}+1\right) x^{2}-4 \sqrt{2} k x=0$
$\Rightarrow x_{P}=\frac{4 \sqrt{2} k}{2 k^{2}+1}$.
Thus, point $P\left(\frac{4 \sqrt{2} k}{2 k^{2}+1}, \frac{2 \sqrt{2} k^{2}-\sqrt{2}}{2 k^{2}+1}\right)$.
Similarly, point $Q\left(\frac{2 \sqrt{2} k^{\prime}}{k^{\prime 2}+2}, \frac{\sqrt{2} k^{\prime 2}-2 \sqrt{2}}{k^{\prime 2}+2}\right)$.
Since $k^{\prime}=4 k$, we have
point $Q\left(\frac{4 \sqrt{2} k}{8 k^{2}+1}, \frac{8 \sqrt{2} k^{2}-\sqrt{2}}{8 k^{2}+1}\right)$.
Then $k_{P Q}=\frac{\frac{8 \sqrt{2} k^{2}-\sqrt{2}}{8 k^{2}+1}-\frac{2 \sqrt{2} k^{2}-\sqrt{2}}{2 k^{2}+1}}{\frac{4 \sqrt{2} k}{8 k^{2}+1}-\frac{4 \sqrt{2} k}{2 k^{2}+1}}=-\frac{1}{2 k}$.
Therefore, $l_{P Q}: y-\frac{2 \sqrt{2} k^{2}-\sqrt{2}}{2 k^{2}+1}=-\frac{1}{2 k}\left(x-\frac{4 \sqrt{2} k}{2 k^{2}+1}\right)$
$\Rightarrow y=-\frac{1}{2 k} x+\sqrt{2}$
$\Rightarrow$ line $P Q$ passes through the fixed point $(0, \sqrt{2})$.