The equation of the curve is \(y = \frac{k}{x}\). To find the gradient, we differentiate \(y\) with respect to \(x\).

\(\frac{dy}{dx} = -\frac{k}{x^2}\).

We are given that the gradient \(\frac{dy}{dx} = -3\) when \(x = 2\).

Substitute \(x = 2\) into the derivative:

\(-\frac{k}{2^2} = -3\).

Simplify:

\(-\frac{k}{4} = -3\).

Multiply both sides by \(-4\):

\(k = 12\).