(i) Calculate the distance AB:

\(AB = \sqrt{(5 - (-3))^2 + (1 - 7)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = 10\)

Calculate the distance BC:

\(BC = \sqrt{(5 - (-1))^2 + (1 - k)^2} = \sqrt{6^2 + (k - 1)^2}\)

Given AB = BC, set the equations equal:

\(10 = \sqrt{36 + (k - 1)^2}\)

Square both sides:

\(100 = 36 + (k - 1)^2\)

Simplify:

\(64 = (k - 1)^2\)

Take the square root:

\(k - 1 = \pm 8\)

So, \(k = 9\) or \(k = -7\).

(ii) Find the midpoint M of AB:

\(M = \left( \frac{-3 + 5}{2}, \frac{7 + 1}{2} \right) = (1, 4)\)

Calculate the slope of AB:

\(m_{AB} = \frac{1 - 7}{5 - (-3)} = \frac{-6}{8} = -\frac{3}{4}\)

The slope of the perpendicular bisector is the negative reciprocal:

\(m_{\text{perp}} = \frac{4}{3}\)

Equation of the perpendicular bisector:

\(y - 4 = \frac{4}{3}(x - 1)\)

Set \(y = 0\) to find the x-intercept:

\(0 - 4 = \frac{4}{3}(x - 1)\) \(-4 = \frac{4}{3}(x - 1)\) \(-12 = 4(x - 1)\) \(-12 = 4x - 4\) \(-8 = 4x\) \(x = -2\)

Thus, the coordinates of D are \((-2, 0)\).