Locus of Points on a Segment (Ellipse)

This applet demonstrates the locus formed by dividing points on a line segment as the segment moves in a plane. Through interactive manipulation of the segment's endpoints or the division ratio, users can observe that the collection of division points traces out an ellipse. The construction begins with a line segment whose endpoints move along two fixed perpendicular lines, or alternatively, one endpoint slides along a circle while the other remains fixed. A point divides the segment in a constant ratio, and as the segment undergoes its prescribed motion, the dividing point generates a smooth closed curve. By adjusting the slider for the division ratio or dragging the endpoints directly, students can explore how different ratios produce ellipses of varying shapes and orientations, including the special case where the ratio equals one-half, yielding a circle. The applet seamlessly integrates analytic geometry and synthetic geometric construction. On the analytic side, parametric equations describe the coordinates of the dividing point as functions of the moving angle or parameter, revealing the standard form of an ellipse. On the geometric side, the construction visually shows how the locus emerges from simple linear motion constrained by the division ratio. This dual perspective helps learners connect the algebraic representation x equals a times cosine theta and y equals b times sine theta to the physical act of tracing a point on a moving segment, deepening their understanding of parametric equations and their geometric significance. Students benefit from this interactive exploration in several ways. First, they gain an intuitive grasp of what an ellipse is beyond the familiar focus-directrix or string-and-pins definitions, seeing it as a natural locus generated by affine motion. Second, they can experiment dynamically, immediately observing how changes to parameters affect the resulting curve, which reinforces the relationship between algebraic coefficients and geometric shape. Third, the visual linkage between parametric description and geometric construction strengthens conceptual understanding that is often difficult to achieve through static diagrams or symbolic manipulation alone. The applet serves as a valuable pedagogical tool for courses in analytic geometry, calculus, or precalculus, where the ellipse as a locus and its parametric representation are key topics.