A conditional extremum problem of many variable functions is usually solved by method of Lagrange multipliers. In this paper, the authors summarized some conditional extremum problems of two variable functions from an initial problem. Firstly, the authors also used method of Lagrange multipliers to find extremums of given functions and maximum or minimum values of these functions. Then, acknowledging these problems at geometric aspect in two-dimensional space, that is the shortest distance or the longest distance from a point to a line, a circle or an ellipse.