Locus: Equidistant Point from a Fixed Point and a Fixed Line (Parabola)

This applet demonstrates the geometric definition of a parabola through an interactive locus construction. It begins with a fixed point, known as the focus, and a fixed line, known as the directrix, placed in the coordinate plane. A moving point is then constructed so that its distance to the focus always equals its perpendicular distance to the directrix. As the moving point traces out all possible positions satisfying this condition, the applet dynamically reveals the familiar U-shaped curve of a parabola. The construction is built step by step to make the underlying geometry transparent. At each position of the moving point, the applet displays the segment connecting the point to the focus and the perpendicular segment dropping from the point to the directrix. These two segments are shown to be equal in length at every instant, visually reinforcing the defining property of the parabola. The trajectory feature in GeoGebra captures the path of the moving point and draws the complete curve in real time. Users can interact with the applet in several meaningful ways. They can drag the focus to different locations, move or rotate the directrix, and adjust numerical parameters through input boxes. Each modification causes the parabola to change shape, orientation, and position accordingly. For example, moving the focus farther from the directrix produces a wider parabola, while bringing them closer together creates a narrower one. Rotating the directrix tilts the axis of symmetry of the parabola, helping students see that the focus-directrix relationship holds regardless of orientation. The applet also highlights the axis of symmetry and the vertex of the parabola, which lies exactly midway between the focus and the directrix. These features are labeled and can be explored further as users manipulate the configuration. This interactive visualization is particularly valuable for students learning conic sections. By actively engaging with the construction rather than passively observing a static diagram, students develop a deeper intuitive understanding of why a parabola has the shape it does and how its key elements are related. The immediate visual feedback from adjusting parameters encourages experimentation and helps solidify the connection between the algebraic equation of a parabola and its geometric definition. The applet is well suited for classroom demonstrations, independent exploration, and formative assessment in secondary mathematics courses covering analytic geometry and conic sections.