Let C< subset of>P-r be a linearly normal projective integral curve of arithmetic genus g1 and degree d=2g+1+p for some p1. It is well known that C is cut out by quadric and satisfies Green-Lazarsfeld's property N-p. Recently it is known that for any q P-r\C such that the linear projection (q): CPr-1 of C from q is an embedding, the projected image C-q: =(q)(C)< subset of>Pr-1 is 3-regular, and hence its homogeneous ideal is generated by quadratic and cubic equations. In this article we study the problem when C-q is still cut out by quadrics. Our main result in this article shows that if the relative location of q with respect to C is general then the homogeneous ideal of C-q is still generated by quadrics and the syzygies among them are generated by linear syzygies for the first a few steps.