10. (20 points) Through the right focus $F$ of the ellipse $C: \frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, a line is drawn intersecting the ellipse $C$ at points $A$ and $B$. It is known that $|A B|=8$. Find the equation of the line $A B$.

 untranslated part:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 

Note: The part "将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。" is a request and not part of the original text to be translated. It has been translated above as a note.
10. From the problem, we know the point $F(3,0)$. Let $l_{A B}: x=m y+3$.

From $\left\{\begin{array}{l}16 x^{2}+25 y^{2}=400, \\ x=m y+3,\end{array}\right.$ we have
$$
\left(16 m^{2}+25\right) y^{2}+96 m y-256=0 \text {. }
$$

Let points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$. Then
$$
y_{1}+y_{2}=\frac{-96 m}{16 m^{2}+25}, y_{1} y_{2}=\frac{-256}{16 m^{2}+25} \text {. }
$$

Since $|A B|=8$, we have
$$
\begin{array}{l}
\sqrt{1+m^{2}} \cdot \sqrt{\left(y_{1}+y_{2}\right)^{2}-4 y_{1} y_{2}}=8 \\
\Rightarrow\left(1+m^{2}\right)\left[\left(\frac{-96 m}{16 m^{2}+25}\right)^{2}+4 \times \frac{256}{16 m^{2}+25}\right]=64 \\
\Rightarrow m= \pm \frac{\sqrt{5}}{2} .
\end{array}
$$

Thus, $l_{A B}: 2 x \pm \sqrt{5} y-6=0$.