Example 1. In $\triangle A B C$, $\angle C=3 \angle A$, $a=27, c=48$. What is $b=$ ?
(A) 33 ;
(B) 35;
(C) 37 ;
(D) 39 ;
(E) The value of $b$ is not unique.

[36th American High School Mathematics Examination (February 26, 1985, Beijing), Question 28]

Your publication provided a trigonometric solution to this problem in the 3rd issue of 1985. Now, we will use the aforementioned necessary and sufficient condition to solve it.

保留源文本的换行和格式，直接输出翻译结果。
Solve: In $\angle BCD$, construct $\angle BCD = \angle A$, then $\angle ACD = 2\angle A$.
Let $AD = x, CD = y$. Then
$$
\begin{array}{l}
x^{2} - y^{2} = yb, \\
\because \triangle ACB \sim \triangle CDB, \\
\therefore \frac{y}{b} = \frac{27}{48} = \frac{9}{16}, y = \frac{9}{16} b \\
\frac{48 - x}{27} = \frac{27}{48}, \\
x = \frac{75 \times 21}{48} = \frac{25 \times 21}{16}.
\end{array}
$$
(2) Substituting
(3) into
(1) gives
$$
\left(\frac{25 \times 21}{16}\right)^{2} - \left(\frac{9}{16}\right)^{2} b^{2} = \frac{9}{16} b^{2}.
$$

Solving for $b$ yields $b = 35$, hence (B) is true.