2021 THREE(d)

A curve has the equation $y=\frac{4x+k}{4x-k}$, where $k$ is a constant and $x\ne \frac{k}{4}$.

The point P lies on the curve and has an $x$-coordinate of 3.

The gradient of the tangent to the curve at P is $\frac{-8}{27}$.

Find the possible value(s) of $k$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*