(i) To find the coordinates of \(X\), we first calculate the slope of \(AB\):

\(m_{AB} = \frac{6 - 14}{14 - 2} = -\frac{2}{3}\).

The slope of \(CX\), being perpendicular to \(AB\), is the negative reciprocal:

\(m_{CX} = \frac{3}{2}\).

The equation of line \(AB\) is:

\(y - 14 = -\frac{2}{3}(x - 2)\).

The equation of line \(CX\) is:

\(y - 2 = \frac{3}{2}(x - 7)\).

Solving these equations simultaneously gives the coordinates of \(X\):

\(X (11, 8)\).

(ii) To find the ratio \(AX : XB\), calculate the distances:

\(AX = \sqrt{(11 - 2)^2 + (8 - 14)^2} = \sqrt{9^2 + 6^2} = \sqrt{117}\).

\(XB = \sqrt{(14 - 11)^2 + (6 - 8)^2} = \sqrt{3^2 + 2^2} = \sqrt{13}\).

The ratio \(AX : XB = \sqrt{117} : \sqrt{13} = 3:1\).