LATTICE POINTS ON THE HOMOGENEOUS CONE z2 = 10x2 ? 6y2

The Cone represented by the homogeneous ternary quadratic equation given by is considered for determining non-zero distinct integral points on it. Four different patterns of solutions, each satisfying the cone under consideration are illustrated. A few interesting properties between the integral solutions on the cone and special number patterns namely,polygonal numbers, star numbers, woodall numbers , pentatope numbers are exhibited. Also, knowing an integral point on the given cone, three different triples of the non-zero distinct integers , each satisfying the given homogeneous cone are obtained.