10. (20 points) Given the equation of curve $C$ as
$$
\frac{\left(16 \lambda^{2}-9\right) x^{2}}{112}+\frac{\lambda^{2} y^{2}}{7}=1(|x| \leqslant 4) \text {. }
$$
(1) For $\lambda>0$, discuss the shape of curve $C$;
(2) Let $M$ be a moving point on curve $C$, and $P$ be a point on the line passing through $M$ and perpendicular to the $x$-axis, such that $\frac{|O P|}{|O M|}=\lambda$, find the equation of the trajectory of point $P$.
10. (1) Discuss in four cases (see Fig. 4).
(i) When $0<\lambda<\frac{3}{4}$, the curve
$C$ is
$$
\frac{x^{2}}{\frac{112}{16 \lambda^{2}-9}}+\frac{\lambda^{2} y^{2}}{7}=1(|x| \leqslant 4),
$$

This is the part of an ellipse centered at the origin with the major axis on the $x$-axis, satisfying $|x| \leqslant 4$.
(ii) When $\lambda=\frac{3}{4}$, the curve $C$ is
$$
\frac{x^{2}}{16}+\frac{\lambda^{2} y^{2}}{7}=1(|x| \leqslant 4),
$$

This is the part of an ellipse centered at the origin with the major axis on the $x$-axis, satisfying $|x| \leqslant 4$.
(iii) When $\frac{3}{4}<\lambda<1$, the curve $C$ is
$$
\frac{x^{2}}{\frac{112}{16 \lambda^{2}-9}}+\frac{\lambda^{2} y^{2}}{7}=1(|x| \leqslant 4),
$$

This is the part of an ellipse centered at the origin with the major axis on the $x$-axis, satisfying $|x| \leqslant 4$.
(iv) When $\lambda \geqslant 1$, $\frac{112}{16 \lambda^{2}-9} \leqslant 16$, the curve $C$ is
$$
\frac{x^{2}}{\frac{112}{16 \lambda^{2}-9}}+\frac{\lambda^{2} y^{2}}{7}=1,
$$

This is an ellipse centered at the origin with the major axis on the $x$-axis.
(2) Let $M\left(x_{0}, y_{0}\right) 、 P(x, y)$, where $x_{0}=x$ $\in[-4,4]$.
By $\frac{|O P|}{|O M|}=\lambda$, we have (see Fig. 4)
$$
\frac{x^{2}+y^{2}}{x_{0}^{2}+y_{0}^{2}}=\lambda^{2} \text {, }
$$

Thus, $y_{0}^{2}=\frac{\left(1-\lambda^{2}\right) x^{2}+y^{2}}{\lambda^{2}}$.
Since point $M$ is on the curve $C$, we have
$$
\frac{\left(16 \lambda^{2}-9\right) x^{2}}{112}+\frac{\lambda^{2}}{7} \cdot \frac{\left(1-\lambda^{2}\right) x^{2}+y^{2}}{\lambda^{2}}=1 \text {, }
$$

which simplifies to $\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$.
Therefore, the trajectory of point $P$ is the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$.