1. Find the gradient of line \(AB\):

\(\text{Gradient of } AB = \frac{5h - h}{4h + 6 - h} = \frac{4h}{3h + 6}\)

2. The gradient of the perpendicular bisector is \(-\frac{3}{2}\), so:

\(\frac{4h}{3h + 6} \times \left(-\frac{3}{2}\right) = -1\)

3. Solve for \(h\):

\(\frac{4h}{3h + 6} = \frac{2}{3}\)

\(12h = 2(3h + 6)\)

\(12h = 6h + 12\)

\(6h = 12\)

\(h = 2\)

4. Find the midpoint of \(AB\):

\(\left(\frac{5h + 6}{2}, \frac{3h}{2}\right) = (8, 6)\)

5. Substitute the midpoint into the equation of the bisector:

\(3(8) + 2(6) = k\)

\(24 + 12 = k\)

\(k = 36\)