Because we know that the equation of a circle is $(x-a)^{2}+(y-b)^{2}=r^{2}$ where the center of the circle is $(a, b)$ and the radius is $r$，we can find the equation of this circle by centering it on the origin．Doing this，we get that the equation is $x^{2}+y^{2}=r^{2}$．Now，we can set the distance between the chords as $2 d$ so the distance from the chord with length 38 to the diameter is $d$．
Therefore，the following points are on the circle as the $y$ -axis splits the chord in half，that is where we get our $x$ value：
$(19, d)$
$(19,-d)$
$(17,-3 d)$
Now，we can plug one of the first two value in as well as the last one to get the following equations：
$$
\begin{gathered}
19^{2}+d^{2}=r^{2} \\
17^{2}+(3 d)^{2}=r^{2}
\end{gathered}
$$
Subtracting these two equations，we get $19^{2}-17^{2}=8 d^{2}$ - therefore，we get $72=8 d^{2} \rightarrow d^{2}=9 \rightarrow d=3$．We want to find $2 d=6$ because that's the distance between two chords．So，our answer is $B$．