Answer: k = 6
The line bisects the segment, so the midpoint of the segment lies on the line.
Midpoint \(= \left(\dfrac{8 + 0}{2}, \dfrac{-1 + k}{2}\right) = \left(4, \dfrac{k - 1}{2}\right)\).
Substituting in \(x - 6y + 11 = 0\): \(4 - 6\left(\dfrac{k - 1}{2}\right) + 11 = 0\), i.e. \(15 - 3(k - 1) = 0\).
\(3k = 18\), so \(k = 6\).