Let us take a look at the points at which the curve cuts the X axis for nonzero r. These are the points with Cartesian coordinates (-2,0) and (-4,0), respectively.
One of them correspond to t=0, the other to t=pi. The r values for these two points must be 1-k and 1+k, respectively. Of these, the first must be negative (a positive r for t=0 would lead to a point to the right of the origin), leading to a distance from the origin of k-1. Since this is smaller than k+1, this must correspond to
(-2,0)
(The above follows simply from the correspondence x = r cos(t), y = r sin(t) between polar and Cartesian coordinates.)
Thus
-2=1-k cos(0) = 1-k
This will lead to k =3
A check : note that this is consistent with r(pi)=4 - the other point on the X axis.