Ellipse Locus Construction from Two Concentric Circles

This applet demonstrates how to construct the locus of an ellipse using two concentric circles and geometric constructions. By selecting points on the two concentric circles, drawing corresponding perpendicular lines, and finding their intersections, the applet visually illustrates the geometric meaning and generation process behind the parametric equations of an ellipse. The construction reveals that if the larger circle has radius equal to the ellipse's semi-major axis and the smaller circle has radius equal to the semi-minor axis, the intersection point traces out an ellipse as the selected points rotate around the circles. This elegant method directly connects the parametric form x = a cos(t), y = b sin(t) to a concrete geometric procedure, helping students understand where these equations come from rather than treating them as abstract formulas. The interactive nature of the applet allows learners to drag points, adjust the radii of the concentric circles, and observe in real time how the ellipse shape changes. This makes it an effective teaching tool for coordinate geometry and parametric curves, enabling students to discover the relationship between circles and ellipses through direct manipulation and visual exploration.