10. Point $P\left(x_{0}, y_{0}\right)$ is any point on the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{2}=1$, and the line $y=k x+m(k m \neq 0)$ intersects the ellipse $C_{1}: \frac{x^{2}}{4}+y^{2}=1$ to form a chord $M N$ which is bisected by the line $O P$ and satisfies $m y_{0}>-1$. The area of $\triangle P M N$ is 1. Determine the shape of quadrilateral $O M P N$ and the number of common points between the line $P M$ and the ellipse $C_{1}$. Prove your conclusions.
10. Analysis: Substituting \( y = kx + m \) into \( \frac{x^2}{4} + y^2 = 1 \), we get: \( \left(4k^2 + 1\right)x^2 + 8kmx + 4m^2 - 4 = 0 \), hence \( x_1 + x_2 = -\frac{8km}{4k^2 + 1}, x_1 x_2 = \frac{4m^2 - 4}{4k^2 + 1} \), then
$$
|MN| = \sqrt{k^2 + 1} |x_1 - x_2| = \sqrt{k^2 + 1} \sqrt{\frac{64k^2m^2}{(4k^2 + 1)^2} - \frac{16m^2 - 16}{4k^2 + 1}} = \frac{4 \sqrt{k^2 + 1}}{4k^2 + 1} \sqrt{4k^2 + 1 - m^2}
$$

Let \( M(x_1, y_1), N(x_2, y_2) \). Since \( MN \) is bisected by the line \( OP \), by the point difference method, we get: \( k = -\frac{x_0}{4y_0} \).
Since \( S_{\triangle PMN} = \frac{1}{2} |MN| d_P = \frac{2|kx_0 + m - y_0|}{4k^2 + 1} \sqrt{4k^2 + 1 - m^2} = \frac{2|m - \frac{2}{y_0}|}{\frac{2}{y_0^2}} \sqrt{\frac{2}{y_0^2} - m^2} = 1 \)
Simplifying, we get: \( (2 - m^2 y_0^2)(2 - m y_0)^2 = 1 \). Let \( m y_0 = t \), then \( (t - 1)(t^3 - 3t^2 - t + 7) = 0 \); \( \triangle > 0 \), we get: \( \frac{2}{y_0^2} - m^2 > 0 \), i.e., \( t^2 - 1 \), hence \( t^3 - 3t^2 - t + 7 = (t - 3)(t^2 - 2) + t + 1 > 0 \), then \( t = 1 \), i.e., \( m y_0 = 1 \). Let the distance from \( O \) to \( MN \) be \( d' \), then
$$
d_P = \frac{|kx_0 + m - y_0|}{\sqrt{k^2 + 1}} = \frac{|-\frac{2}{y_0} + m|}{\sqrt{k^2 + 1}} = \frac{|m|}{\sqrt{k^2 + 1}} = d'
$$

Therefore, \( OP \) is bisected by the line \( MN \), so the quadrilateral \( OMPN \) is a parallelogram, so \( x_0 = x_1 + x_2, y_0 = y_1 + y_2 \) so \( x_0^2 + 4y_0^2 = (x_1 + x_2)^2 + 4(y_1 + y_2)^2 = 8 + 2x_1 x_2 + 8y_1 y_2 = 8 \), so \( x_1 x_2 + 4y_1 y_2 = 0 \) i.e., \( k_{OM} k_{ON} = -\frac{1}{4} \), i.e., \( k_{PM} = k_{ON} = -\frac{x_1}{4y_1} \), so the equation of the line \( PM \) is \( \frac{xx_1}{4} + yy_1 = 1 \) which is the tangent line to the ellipse \( C_1 \) at \( M \), therefore the number of common points between the line \( PM \) and the ellipse \( C_1 \) is 1.