A study was conducted to compute the distance from an ellipsoid to a linear surface and a quadric in Rⁿ¹ and to investigate the closeness of objects in an n-dimensional parametric space in computational geometry and pattern recognition problem. The study found that the optimization problem of these is reduced by traditional application of LaGrange multipliers to a problem of solving a system of algebraic equations. The study stated that eliminate all variables from the system and construct an algebraic equation to evaluate the coordinates of the nearest points. The coordinates of the nearest points on the given surface can be expressed as a rational function in terms of this value. The study found that the procedure used is implemented by constructing a Gröbner basis and the methods of classical elimination theory.