Step-by-step Solution:

1. Substitute point A (8, -4) into the line equation 4x + ky = 20:
   \(4(8) + k(-4) = 20\)
   \(32 - 4k = 20\)
   Solve for \(k\):
   \(32 - 20 = 4k\)
   \(12 = 4k\)
   \(k = 3\)
2. Substitute point B (b, 2b) into the line equation:
   \(4b + 3(2b) = 20\)
   \(4b + 6b = 20\)
   \(10b = 20\)
   Solve for \(b\):
   \(b = 2\)
3. Find the mid-point of AB:
   Mid-point formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
   \(A (8, -4), B (2, 4)\)
   Mid-point: \(\left( \frac{8 + 2}{2}, \frac{-4 + 4}{2} \right) = (5, 0)\)