Example 1. Let $G$ be any convex shape tightly sandwiched between two parallel lines $L_{1}$ and $L_{2}$, with its boundary $C$ intersecting both $L_{1}$ and $L_{2}$. A line $L$ parallel to $L_{1}$ divides $G$ into two parts, $A$ and $B$, as shown in the figure, and the distances between $L$ and $L_{1}$, and $L$ and $L_{2}$ are $a$ and $b$, respectively.
(1) For what shape of $G$ is the ratio of the areas of $A$ and $B$, $\frac{S_{A}}{S_{B}}$, maximized? Explain your reasoning.
(2) Find the maximum value of $\frac{S_{A}}{S_{B}}$.
Let $M$ be one of the intersection points of $G$ and $L_{2}$, and the intersection of $L$ and $G$ must be a line segment, which we can denote as $N P$. To maximize $\frac{S_{A}}{S_{B}}$, $S_{A}$ should be as large as possible, and $S_{B}$ should be as small as possible.

Connect $M N, M P$. Clearly, $\triangle M N P$ is contained within the convex shape $G$, so $S_{B} \geqslant S_{\triangle M N P}$.

Extend $M N, M P$ to intersect $L_{1}$ at points $Q, R$ respectively. It can be proven that $S_{A}$ is contained within the trapezoid $N P R Q$, (assuming otherwise, there exists a point $Y$ in $S_{A}$ outside the trapezoid $N P R Q$, connect $M X$, it intersects $L$ at point $Y$, clearly $Y$ is outside the line segment $N P$, a contradiction). Therefore, $S_{A} \leqslant S_{\text {trapezoid } NPRQ}$.

Thus, the maximum value of $\frac{S}{3}$ is $\frac{a^{2}+2 a b}{b^{2}}$, at this point $G$ is a triangle with one side on $L_{1}$ and another side on $L_{2}$.

If a line $L$ intersects the convex shape $G$ and one side of $L$ contains no points of $G$, then $L$ is called a supporting line of $G$. Examining supporting lines is an effective method for handling convex shape problems.