In this paper we propose a new local quasi-Newton method to solve the equality constrained non-linear programming problem. The pivotal feature of the algorithm is that a projection of the Hessian of the Lagrangian is approximated by a sequence of symmetric positive definitive matrices. The matrix approximation is updated at every iteration by a projected version of the DFP or BFGS formula. We establish that the method is locally convergent and the sequence of x-values converges to the solution at a 2-step Q-superlinear rate.