Abstarct: Traditionally, the minisum and minimax circle location problems concern with finding a circle C in the plane such that the sum of the distances and maximum distance from the circumference of the circle to the given points is minimized, respectively. The radius of the circle can be fixed or variable. In this thesis we consider the inverse case, that is: a circle C is given and we want to modify the coordinate or weight of existing points with the minimum cost such that the given circle be optimal. Mathematical models and some properties of the cases that the circle C becomes optimal with comparing to all other circles, and the circle C becomes the best circle with comparing to the circles with radius r0 are presented. In addition to we investigate semi-obnoxious circle problem. When the facilities are desirable to a part of clients and undesirable to the rest of them, we face these kind of location problems. In this paper, we consider the minisum circle location problem with positive and negative weights using Euclidean norm which is a nonlinear optimization problem. We show that some properties of the minisum circle location problem with positive weights, also hold for the semi-obnoxious case