Example 2. Find a point $M$ on the hyperbola $\frac{x^{2}}{3} -\frac{y^{2}}{1}=1$ such that the sum of its distances to points $A(2,0)$ and $B$ $(8,2 \sqrt{2})$ is minimized.
Solve: As shown in Figure 2, $A(2,0), A^{\prime}(-2,0)$ are the foci. Connect $A^{\prime} B$, intersecting the hyperbola (right branch) at $M$, then $M(3$, $\sqrt{2}$ ) is the point required, and the minimum value is $4 \sqrt{3}$.