6. 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 $3\left|P F_{1}\right|=4\left|Q F_{1}\right|$, then the ratio of the minor axis to the major axis of the ellipse $\Gamma$ is $\qquad$
6. $\frac{2 \sqrt{6}}{7}$.

Assume $\left|P F_{1}\right|=4,\left|Q F_{1}\right|=3$.
Let the lengths of the major axis and minor axis of the ellipse $\Gamma$ be $2 a$ and $2 b$, respectively, and the focal distance be $2 c$.
Then $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|=2 c$, and by the definition of an ellipse, $2 a=\left|Q F_{1}\right|+\left|Q F_{2}\right|=\left|P F_{1}\right|=\left|P F_{2}\right|=2 c+4$.
Thus, $\left|Q F_{2}\right|=\left|P F_{1}\right|+\left|P F_{2}\right|-\left|Q F_{1}\right|=2 c+1$.
As shown in Figure 3, let $H$ be the midpoint of segment
$P F_{1}$. Then
$$
\left|F_{1} H\right|=2,|Q H|=5,
$$

and $F_{2} H \perp P F_{1}$.
By the Pythagorean theorem, we have
$$
\begin{array}{l}
\left|Q F_{2}\right|^{2}-|Q H|^{2} \\
=\left|F_{2} H\right|^{2} \\
=\left|F_{1} F_{2}\right|^{2}-\left|F_{1} H\right|^{2} \\
\Rightarrow(2 c+1)^{2}-5^{2}=(2 c)^{2}-2^{2} \\
\Rightarrow c=5 \Rightarrow a=7, b=2 \sqrt{6} \\
\Rightarrow \frac{2 b}{2 a}=\frac{2 \sqrt{6}}{7} .
\end{array}
$$