Locus of Conic Sections: Center of Circle Tangent to a Fixed Circle and Passing

This GeoGebra applet demonstrates the geometric definition and locus generation of conic sections through an elegant and interactive dynamic construction. By constructing a moving circle that passes through a fixed point A and is tangent to a fixed circle, the applet explores the continuous trajectory of the moving circle's center. When point A is positioned inside the fixed circle, the locus of the moving circle's center forms an ellipse. Conversely, when point A is dragged outside the fixed circle, the locus seamlessly transitions to form a hyperbola. Users can actively interact with the model by dragging point A across the boundary of the fixed circle to observe this real-time geometric transformation. Additionally, learners can adjust the radius of the fixed circle or reposition its center to investigate how these parameters dictate the shape, size, and eccentricity of the resulting conic section. The applet also allows users to trace the path of the moving circle's center, leaving a clear visual trail of the generated locus. Mathematically, this visualization is rooted in the fundamental focal definitions of conic sections. Let the fixed circle have center O and radius R, and let the moving circle have center P. When A is inside the fixed circle, the moving circle is internally tangent, meaning the distance OP plus the distance AP equals the constant radius R. This constant sum of distances to two fixed points, O and A, is the exact geometric definition of an ellipse. When A is outside, the tangency condition yields a constant absolute difference between the distances OP and AP, which perfectly defines a hyperbola with foci at O and A. Students benefit immensely from this interactive approach, as it bridges the gap between synthetic geometry and analytic algebra. Instead of merely memorizing abstract equations, learners can visually and kinesthetically experience how conic sections emerge from simple tangency constraints. This deepens their understanding of focal properties, enhances spatial reasoning, and provides a robust geometric intuition that is highly valuable for advanced studies in calculus, physics, and analytic geometry.