6. If $x^{2}+y^{2}=169$, then the function
$$
\begin{aligned}
f(x, y)= & \sqrt{24 y-10 x+338}+ \\
& \sqrt{24 y+10 x+338}
\end{aligned}
$$

has the maximum value of ( ).
(A) $10 \sqrt{26}$
(B) $13 \sqrt{2}$
(C) $5 \sqrt{26}$
(D) $26 \sqrt{2}$
6.A.

From $x^{2}+y^{2}=169$, we know
$[f(x, y)]^{2}=48 y+676+2 \sqrt{(24 y+338)^{2}-(10 x)^{2}}$.
When $y$ is larger and $|x|$ is smaller, $[f(x, y)]^{2}$ is larger.
Therefore, when $y=13, x=0$, $f(x, y)=10 \sqrt{26}$ is the maximum value.