Example 10 Given that $\triangle ABC$ intersects with a line $PQ$ parallel to $AC$, and the area of $\triangle APQ$ equals a constant $k^2$. What is the relationship between $\iota^2$ and the area $S$ of $\triangle ABC$ for the problem to have a solution? How many solutions are there?
(1987, Shanghai Junior High School Mathematics Competition)
Solution: Let $\frac{B P}{A B}=x$, then
$S_{\triangle B P Q}=x^{2} S, S_{\triangle A B Q}=x S$.
According to the problem, we have $x S-x^{2} S=k^{2}$,
which is $S x^{2}-S x+k^{2}=0$.
The condition for this equation to have real solutions is
$\Delta=S^{2}-4 S k^{2} \geqslant 0$, that is, $S \geqslant 4 k^{2}$.
Moreover, the two solutions $x_{1} 、 x_{2}$ satisfy
$$
\begin{array}{l}
x_{1}+x_{2}=1,04 k^{2}$ when, this problem has two solutions; when $S=$ $4 k^{2}$, this problem has one solution.

Note: The key to solving geometric problems using the discriminant is to construct a quadratic equation based on the known conditions and the quantitative characteristics of the figure, and then, by the condition for real roots to exist, solve the problem.
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