Definition and Locus Animation of a Hyperbola

This GeoGebra applet visually demonstrates the mathematical definition of a hyperbola through dynamic geometric construction. Utilizing two fixed foci labeled F_1 and F_2, and the defining geometric property that the absolute difference of distances from any point on the hyperbola to the two foci remains constant, the applet generates the hyperbola locus step by step. The construction proceeds by drawing line segments connecting a moving point to each focus, constructing the perpendicular bisector of one segment, locating the intersection points between relevant lines and bisectors, and then applying the Locus command to trace the complete hyperbola as the parameter varies. The animation clearly shows how each intermediate geometric element contributes to the final curve, reinforcing the connection between the algebraic definition and its visual realization. This interactive demonstration helps students deepen their understanding of how a hyperbola is formed, its key geometric features such as the transverse axis, the relationship between the constant difference and the focal distance, and the symmetric structure of both branches. By manipulating the positions of the foci or adjusting the constant difference parameter, learners can observe in real time how these changes affect the shape and orientation of the hyperbola, making abstract properties concrete and intuitive. The applet is particularly valuable for visual learners and for classroom instruction on conic sections, providing a dynamic bridge between formal definitions and geometric intuition.