This video shows how to check the constraint qualification for a nonlinear constrained optimization problem and what might happen, if the constraint qualification is not satisfied.
I assumed all three to be binding (holding with equality around the 4 min mark), which implies c1=0 and c2=0 and k=p1*0+p2*0=0. When I only require c1=0 and c2=0 (around the 8 min mark), I do not require that k=0.