Example 1 Let the two foci of the ellipse $\Gamma$ be $F_{1}$ and $F_{2}$, and a line passing through point $F_{1}$ intersects the ellipse $\Gamma$ at points $P$ and $Q$. If $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|$, and $3\left|P F_{1}\right|=4\left|Q F_{2}\right|$, then the ratio of the minor axis to the major axis of the ellipse $\Gamma$ is $\qquad$. [1]
(2014, National High School Mathematics Joint Competition)
According to the first definition of an ellipse, we have
$$
\left|P F_{1}\right|+\left|P F_{2}\right|=2 a \text {. }
$$

Given that $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|=2 c$, it follows that $\left|P F_{1}\right|=2 a-2 c$.
Let $\angle P F_{1} F_{2}=\theta$.
Thus, the polar equation of the ellipse can be written as
$$
\begin{array}{l}
\rho=\frac{e p}{1-e \cos \theta} . \\
\text { Then }\left|P F_{1}\right|=\frac{e p}{1-e \cos \theta}, \\
\left|Q F_{2}\right|=\frac{e p}{1-e \cos (\pi+\theta)}=\frac{e p}{1+e \cos \theta} .
\end{array}
$$

From $3\left|P F_{1}\right|=4\left|Q F_{2}\right|$, we get
$$
\begin{array}{l}
\frac{3 e p}{1-e \cos \theta}=\frac{4 e p}{1+e \cos \theta} \\
\Rightarrow e \cos \theta=\frac{1}{7} .
\end{array}
$$

Since $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|$, we have
$$
\begin{array}{l}
\left|P F_{1}\right|=2\left|F_{1} F_{2}\right| \cos \theta \\
\Rightarrow 2 a-2 c=4 c \cos \theta \\
\Rightarrow a-c=2 c \cos \theta \\
\Rightarrow 1-e=2 e \cos \theta
\end{array}
$$

From equations (1) and (2), we get $e=\frac{5}{7}$.
Therefore, the ratio of the minor axis to the major axis of the ellipse $\Gamma$ is
$$
\frac{2 b}{2 a}=\frac{b}{a}=\sqrt{1-e^{2}}=\frac{2 \sqrt{6}}{7} .
$$