\(9x^2 - 16y^2 = 144 \Rightarrow \frac{x^2}{16} - \frac{y^2}{9} = 1\), so \(a^2 = 16\) and \(b^2 = 9\).

The line is \(y = -\frac{3}{4}x - \frac{k}{4}\), so \(m = -\frac{3}{4}\) and \(c = -\frac{k}{4}\).

Tangency condition: \(c^2 = a^2m^2 - b^2 = 16 \times \frac{9}{16} - 9 = 0\). So \(c = 0\) and \(k = 0\).

(Note: \(k = 0\) gives \(3x + 4y = 0\), which is actually an asymptote of the hyperbola, a "tangent at infinity". Among the options, 0 is the value that satisfies the tangency condition.)

Answer: (A) 0