(i) To find the value of \(k\), we equate the line and the curve:

\(\frac{x^2}{4} = \frac{x}{k} + k\)

Rearranging gives:

\(kx^2 - 4x - 4k^2 = 0\)

For the line to be tangent, the discriminant must be zero:

\(b^2 - 4ac = 0\)

Substituting \(a = k\), \(b = -4\), \(c = -4k^2\), we get:

\((-4)^2 - 4(k)(-4k^2) = 0\)

\(16 + 16k^3 = 0\)

\(k^3 = -1\)

\(k = -1\)

(ii) Substitute \(k = -1\) into the line equation:

\(y = -x - 1\)

Substitute into the curve equation \(4y = x^2\):

\(4(-x - 1) = x^2\)

\(x^2 + 4x + 4 = 0\)

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

\(x = -2\)

Substitute \(x = -2\) back into \(y = -x - 1\):

\(y = -(-2) - 1 = 1\)

Thus, the coordinates of \(P\) are \((-2, 1)\).