An investigation into the support points for which optimal solutions can be got through super convergent line series of quadratic programming problems has been done. The line search algorithm was used to achieve all these. Support points from the response surface were classified into boundary and interior support points. Two illustrative examples of quadratic programming problems were solved using boundary and interior support points. It was verified that support points from the boundary of the response surface yielded optimal solutions that compared favorably with the existing solutions of the illustrative examples. But the solution of the support points from interior of the response surface was far from optimal when compared with existing solutions.