6. Let the two foci of the ellipse $\Gamma$ be $F_{1}, F_{2}$, and a line passing through point $F_{1}$ intersects $\Gamma$ at points $P, Q$. 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 $\qquad$.
Answer: $\frac{2 \sqrt{6}}{7}$.
Solution: Let's 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 $2 a, 2 b$, 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,
$$
\left|Q F_{2}\right|=\left|P F_{1}\right|+\left|P F_{2}\right|-\left|Q F_{1}\right|=2 c+1 \text {. }
$$

Let $H$ be the midpoint of the line 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,
$$
\left|Q F_{2}\right|^{2}-|Q H|^{2}=\left|F_{2} H\right|^{2}=\left|F_{1} F_{2}\right|^{2}-\left|F_{1} H\right|^{2},
$$

which is $(2 c+1)^{2}-5^{2}=(2 c)^{2}-2^{2}$, solving this gives $c=5$, and thus $a=7$, $b=2 \sqrt{6}$, so the ratio of the minor axis to the major axis of the ellipse $\Gamma$ is $\frac{b}{a}=\frac{2 \sqrt{6}}{7}$.