Answer: B

Video solution:

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Written solution:

The two chords of length $38$ are equally far from the center of the circle. They cannot lie on the two outer lines: then the center would lie on the middle line, whose chord would be a diameter longer than $38,$ not the given $34.$ Therefore, the equal chords lie on adjacent lines, with the center halfway between them. Let that half-distance be $d.$ Then each $38$-chord is distance $d$ from the center, and the $34$-chord is distance $3d$ from the center.

If the circle has radius $r,$ then

$$r^2=19^2+d^2=17^2+(3d)^2.$$

Thus $19^2-17^2=8d^2,$ so $72=8d^2,$ and $d=3.$ The distance between adjacent parallel lines is $2d=6.$

Thus, the answer is B .