Example 6. A line segment is drawn from $P(4,3)$ to any point $Q$ on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, and point $M$ internally divides $P Q$ in the ratio $2: 1$. Find the equation of the locus of $M$, and determine where $M$ is located when the area of the triangle formed by $M$ and the two foci $F_{1}, F_{2}$ of the ellipse is maximized.
Given the conditions, the homothetic center is $P(4,3)$, and the homothetic ratio is $\frac{\lambda}{1+\lambda}=\frac{2}{3}$. By property 5 of homothetic transformations, the relevant data of curve $C'$ are the corresponding data of $C$ multiplied by the homothetic ratio. Therefore, the desired trajectory is an ellipse, with the semi-major and semi-minor axes being $2$ and $\frac{4}{3}$, and the center at $O'\left(\frac{4}{3}, 1\right)$.

To maximize $S_{\triangle M F_{1} F_{2}}$, $M$ should be at the vertex $N\left(\frac{4}{3}, \frac{7}{3}\right)$ of the ellipse where the y-coordinate is the largest.