(i) Finding the coordinates of the intersection points:

Substitute \(y = 9 - \frac{6}{x}\) into \(y + x = 8\):

\(9 - \frac{6}{x} + x = 8\)

Simplify to find:

\(x^2 + x - 6 = 0\)

Solving the quadratic equation:

\((x - 2)(x + 3) = 0\)

Thus, \(x = 2\) or \(x = -3\).

For \(x = 2\), \(y = 9 - \frac{6}{2} = 6\).

For \(x = -3\), \(y = 9 - \frac{6}{-3} = 11\).

So, the points are \((2, 6)\) and \((-3, 11)\).

(ii) Finding the equation of the perpendicular bisector:

Midpoint of \((2, 6)\) and \((-3, 11)\):

\(\left( \frac{2 + (-3)}{2}, \frac{6 + 11}{2} \right) = \left( -\frac{1}{2}, \frac{17}{2} \right)\)

Gradient of the line joining the points:

\(m = \frac{11 - 6}{-3 - 2} = -1\)

Gradient of the perpendicular bisector:

\(m = 1\)

Equation of the perpendicular bisector:

\(y - \frac{17}{2} = 1 \left( x + \frac{1}{2} \right)\)

Simplify to:

\(y = x + 9\)