Example 5 As shown in Figure 3.2.10, given $\triangle A B C$ intersects with a line $D E$ parallel to $B C$, and the area of $\triangle B D E$ equals $k^{2}$, what relationship must exist between $k^{2}$ and $S_{\triangle A B C}$ for the problem to have a solution? How many solutions are there?
Let $\frac{A D}{D B}=\lambda_{1}$, then $\frac{C E}{E A}=\frac{D B}{D A}=\lambda_{3}=\frac{1}{\lambda_{1}}$, and $\frac{B F}{F C}=\lambda_{2}=0$ ( $F$ coincides with $B$), then by Theorem 11, we get
$$
\frac{k^{2}}{S_{\triangle A B C}}=\frac{1}{\left(1+\lambda_{1}\right)\left(I+\frac{1}{\lambda_{1}}\right)}
$$

Rearranging, we get
$$
k^{2} \lambda_{1}^{2}+\left(2 k^{2}-S_{\triangle A B C}\right) \lambda_{1}+k^{2}=0
$$

For this equation to have solutions, it must be that $\triangle=\left(2 k^{2}-S_{\triangle A B C}\right)^{2}-4 k^{2} \cdot k^{2} \geqslant 0$, i.e., $S_{\triangle A B C} \geqslant 4 k^{2}$. Therefore, when $S=4 k^{2}$, the original problem has one solution (at this time $\lambda_{1}=1$, i.e., $D E$ is the midline of $\triangle A B C$); when $S>4 k^{2}$, the original problem has two solutions.