Example 1: A line $l$ is drawn through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ intersecting the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 lines $l$, then $\lambda=$ $\qquad$ (1997 National High School Competition Question)
The answer is 4. The reason is: First, note that for a chord passing through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ and intersecting the right branch at two points, the minimum length is $\frac{2 b^{2}}{a}=4$ if and only if the chord is perpendicular to the $x$-axis. (In fact, in polar coordinates, the equation of the hyperbola can be set as $\rho=\frac{2}{1-\sqrt{3} \cos \theta}$. Let $A\left(\rho_{1}, \theta\right), B=\left(\rho_{2}, \pi+\theta\right)\left(\rho_{1}>0, \rho_{2}>0\right)$, then $|A B|=\frac{2}{1-\sqrt{3} \cos \theta}+\frac{2}{1+\sqrt{3} \cos \theta}=\frac{4}{1-3 \cos ^{2} \theta} \geqslant 4$, with equality when $\theta=\frac{\pi}{2}$.

Second, when there are exactly three lines satisfying the given conditions, there are only two possibilities: (i) Only one line intersects both the left and right branches of the hyperbola, while two lines intersect only the right branch. In this case, the line intersecting both branches must be the $x$-axis, and the distance between its two intersection points is $2 a=2$. However, the length of the two chords intersecting only the right branch, $\lambda>4$, does not meet the given conditions. (ii) Two lines intersect both the left and right branches of the hyperbola, while only one line intersects only the right branch, and this chord must be perpendicular to the $x$-axis (otherwise, by symmetry, there would be two chords intersecting only the right branch). In this case, $|A B|=\lambda=4$, and the lengths of the chords intersecting both branches can also meet this condition. Therefore, $\lambda=4$.