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In this paper, the method of finding the line constrained between two quadratic Bezier curves and also finding the line constrained between a quadratic Bezier curve and a circle is presented. The application of the line constrained can be used in the construction of railway tracks between any obstacle or rolling a ball to the other side of a wall in a way that it just touches the wall. The method used is by using equal root properties of a quadratic equation in order to find one point where the line touches the curve. The work examples involved different curve orientations and different circle positions. Mathematica software is used to compute the solutions for the line constrained and present the solutions graphically. By using the method proposed, the number of intersection points obtained is used to determine the number of lines constrained between two curves and between a curve and a circle. The conclusion on whether all lines are acceptable to be considered as the line constrained are depending on the application of the line. [ABSTRACT FROM AUTHOR]