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.