The midpoint of a line segment joining two points `(x1, y1)` and `(x2, y2)` is given by the formula `((x1 + x2)/2, (y1 + y2)/2)`. In this case, the two points are `(1 - k, - 4)` and `(2, k + 1)`, and the midpoint is given as `(-k , k)`. So, we can write the equations:

`(1 - k + 2)/2 = -k`

`(-4 + k + 1)/2 = k`

Solving the first equation for `k`, we get:

`(3 - k)/2 = -k`

`3 - k = -2k`

`k = -3`

Substituting this value of `k` into the second equation, we get:

`(-4 + k + 1)/2 = k`

`(-4 - 3 + 1)/2 = -3`

`-6/2 = -3`

So, the value of `k` that satisfies both equations is -3