Tangent of Hyperbola and Light Reflection

① This example demonstrates the tangent line at a point on a hyperbola and the path of light rays reflecting off that tangent, illustrating the optical property of a hyperbola. ② Core mathematical concepts include the definition of a hyperbola, tangent equations, focal properties, and the law of reflection (angle of incidence equals angle of reflection), bridging analytic geometry and geometric optics. ③ Construction details: using GeoGebra commands such as Hyperbola to create the curve, Point for a moving point, Vector for direction vectors, Intersect for intersection points, If for conditional logic, and Line for drawing the tangent and reflected rays; objects include foci F, F_1, F_2, tangent line t, moving point A, vector v, and numerous auxiliary lines (b, c, e, i, l, m, etc.) to build the reflection path and interactive button. ④ Suitable for high school or college analytic geometry courses to understand conic optical properties, and for physics demonstrations of reflection laws; ideal for classroom interaction or self-directed exploration.