5. A chord $AB$ is drawn through a focus $F$ of the ellipse $\frac{x^{2}}{6^{2}}+\frac{y^{2}}{2^{2}}=1$. If $|A F|=m,|B F|=n$, then $\frac{1}{m}+\frac{1}{n}=$ $\qquad$
5.3.

As shown in Figure 2, draw $A A_{1}$ perpendicular to the directrix at point $A_{1} . A E \perp x$-axis at point $E$.
$$
\begin{array}{l}
\text { Since } \frac{c}{a}=e=\frac{|F A|}{\left|A A_{1}\right|} \\
=\frac{|F A|}{|D F|+|F E|} \\
=\frac{m}{m \cos \alpha+\frac{b^{2}}{c}} .
\end{array}
$$

Therefore, $\frac{1}{m}=\frac{a-c \cos \alpha}{b^{2}}$.
Similarly, $\frac{1}{n}=\frac{a+c \cos \alpha}{b^{2}}$. Hence $\frac{1}{m}+\frac{1}{n}=\frac{2 a}{b^{2}}=3$.