Locus of the Center of a Circle Tangent to a Parabola and Externally Tangent to

This applet demonstrates how to find the locus equation of the centers of all circles that are tangent to a given parabola and externally tangent to a given circle. Through dynamic construction in GeoGebra, the applet builds a parabola, a fixed circle, and a moving circle that satisfies both tangency conditions simultaneously. Students can drag parameters or points to observe how the moving circle adjusts its position and radius while maintaining tangency with the parabola and external contact with the fixed circle. As the moving circle traces its allowable positions, the path traced by its center—the locus—is displayed in real time, providing an intuitive geometric understanding of an otherwise abstract analytic geometry problem. The applet emphasizes the connection between geometric conditions and algebraic equations. By translating the tangency requirements into algebraic constraints—such as equal distances from the center of the moving circle to the parabola and to the fixed circle—students can derive the exact equation of the locus. This dual geometric-algebraic approach reinforces key concepts in analytic geometry, including the properties of parabolas, circle tangency, distance formulas, and curve parametrization. Interactively, users can manipulate key parameters such as the position and size of the fixed circle or the shape of the parabola, and immediately see how these changes affect the resulting locus. This dynamic exploration helps students develop intuition about how geometric constraints shape algebraic outcomes. The visual feedback makes it easier to verify derived equations and identify special cases or symmetries in the locus. This applet is particularly valuable for students studying conic sections and analytic geometry at the high school or undergraduate level. It serves as both a computational aid and a conceptual bridge between visual geometry and formal algebraic reasoning, making it an effective tool for classroom instruction, self-paced exploration, and homework support.