A parametric version of the ellipsoid method with space scaling by a scalar parameter λ > 0 is considered. For certain values of the parameter λ, it is reduced to the well-known versions of the ellipsoid method, namely, the Shor, Khachiyan, Nemirovsky, and Yudin ones. The properties of two algorithmic implementations of the parameterized ellipsoid method are proved. The first algorithm is based on updating the asymmetric matrix B, and the second one updates the symmetric matrix H = BBT. The applications of the algorithms for solving convex programming problems and finding the saddle point of a convex-concave function are described.