(i) Substitute \(x = \ln 3\) and \(y = \ln 2\) into the equation:

\(6e^{2\ln 3} + ke^{\ln 2} + e^{2\ln 2} = c\).

Simplify using \(e^{2\ln 3} = 9\), \(e^{\ln 2} = 2\), and \(e^{2\ln 2} = 4\):

\(6 \times 9 + k \times 2 + 4 = c\).

\(54 + 2k + 4 = c\).

\(58 + 2k = c\).

(ii) Differentiate the left-hand side with respect to \(x\):

\(\frac{d}{dx}(6e^{2x} + ke^y + e^{2y}) = 0\).

\(12e^{2x} + ke^y \frac{dy}{dx} + 2e^{2y} \frac{dy}{dx} = 0\).

Substitute \(x = \ln 3\), \(y = \ln 2\), and \(\frac{dy}{dx} = -6\):

\(12 \times 9 + 2k(-6) + 2 \times 4(-6) = 0\).

\(108 - 12k - 48 = 0\).

\(60 = 12k\).

\(k = 5\).

Substitute \(k = 5\) into \(58 + 2k = c\):

\(58 + 2 \times 5 = c\).

\(c = 68\).