Example 9: Given $\triangle A B C$ intersects with a line $P Q$ parallel to $A C$ and the area of $\triangle A P Q$ equals a constant $k^{2}$, what is the relationship between $b^{2}$ and the area of $\triangle A B C$? Is there a unique solution?
Given $P Q // A C$, we have $\triangle B P Q \sim \triangle B A C$. Therefore, the area relationship between $\triangle A P Q$ and $\triangle A B C$ can be constructed from the similarity ratio, and the various cases of the equation's solutions can be studied from the discriminant.

Solution: Let $\frac{B P}{A B}=x, S=\triangle A B C$ area, then $S_{\triangle B P Q}=$ $x^{2} S$.

According to the problem, we have $x S - x^{2} S = k^{2}$, which simplifies to $S x^{2} - S x + k^{2} = 0$. Therefore, $\Delta = S^{2} - 4 S k^{2} \geqslant 0$, and $S \geqslant 4 k^{2}$ for the equation to have two solutions. The roots $x_{1}, x_{2}$ of the equation satisfy $x_{1} + x_{2} = 1, 0 < x_{1} x_{2} = \frac{k^{2}}{S} < 1$, so $x_{1}, x_{2}$ are within $(0,1)$. Thus, when $S < 4 k^{2}$, there are two solutions, and when $S = 4 k^{2}$, there is one solution.