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The line location problem, which represents a specific case within the broader class of hyperplane location problems, has attracted considerable research focus location theory. This investigation addresses locating lines from a location science perspective. Given n points situated in the plane, each assigned a positive weight that reflects its relative importance, the median line is defined as the line minimizing the total sum of these weighted distances. Our study is, to our knowledge, the first to examine the inverse median line location problem in the plane under both the Euclidean and rectilinear distance norms. Specifically, when a line L is fixed, the goal is to determine the Minimum-cost modifications to the problem parameters—either the demand point weights or their spatial coordinates—such that L becomes the globally optimal median line. We proceed by developing and analyzing mathematical models that characterize this inverse problem across the two norm settings. We demonstrate that the inverse model, when demand weights are the variables, can be precisely formulated and solved via linear programming. Conversely, for the instance involving necessary modifications to the coordinates, an effective greedy algorithm is proposed for solution approximation. The practical application and performance of this developed methodology are subsequently illustrated through a set of computational experiments.