Hyperbola: Geometric Construction and Intersection

① This example demonstrates the standard geometric definition of a hyperbola: the locus of points where the absolute difference of distances to two fixed foci (F₁, F₂) is constant, visualized through an interactive construction. ② Core mathematical concepts include foci, transverse axis, conjugate axis, asymptotes, and the relationship between intersections and secant lines, all fundamental to conic sections. ③ Construction details: The GeoGebra command Hyperbola generates the curve directly; Point defines foci F₁, F₂ and a moving point C; Segment creates segments r₁ and r₂ representing distances from the moving point to each focus; auxiliary objects M, a, and text1 label key parameters, illustrating the constant distance-difference constraint. ④ Suitable for high school or early college analytic geometry courses, this example helps students understand the definition, standard equation, and geometric properties of hyperbolas, and serves as a dynamic teaching aid for instructors or self-paced exploration by learners.