(i) To find the perpendicular bisector of AB:

1. Calculate the midpoint of AB:

Midpoint = \(\left( \frac{-1 + 7}{2}, \frac{6 + 2}{2} \right) = (3, 4)\)

2. Find the slope of AB:

Slope of AB = \(\frac{2 - 6}{7 + 1} = -\frac{1}{2}\)

3. The slope of the perpendicular bisector is the negative reciprocal:

Slope = 2

4. Use the point-slope form to find the equation:

\(y - 4 = 2(x - 3)\)

\(y = 2x - 6 + 4\)

\(y = 2x - 2\)

(ii) For point C on the perpendicular bisector:

1. The equation of the bisector is \(q = 2p - 2\)

2. The distance OC is 2 units:

\(p^2 + q^2 = 4\)

3. Substitute \(q = 2p - 2\) into \(p^2 + q^2 = 4\):

\(p^2 + (2p - 2)^2 = 4\)

\(p^2 + 4p^2 - 8p + 4 = 4\)

\(5p^2 - 8p = 0\)

\(p(5p - 8) = 0\)

\(p = 0\) or \(p = \frac{8}{5}\)

4. Find q for each p:

If \(p = 0\), \(q = 2(0) - 2 = -2\)

If \(p = \frac{8}{5}\), \(q = 2\left(\frac{8}{5}\right) - 2 = \frac{16}{5} - \frac{10}{5} = \frac{6}{5}\)

5. Possible positions of C are (0, -2) and \(\left( \frac{8}{5}, \frac{6}{5} \right)\).