Lemniscate of Bernoulli: Construction and Drag Test:[br][br]The Lemniscate of Bernoulli is a classical curve of degree four, shaped like a sideways figure eight (∞). It can be defined algebraically by the equation:[br](x² + y²)² = 2a²(x² − y²)[br]It also has a geometric definition: it is the locus of all points P such that the product of the distances to two fixed points (foci) F₁ and F₂ is constant:[br]PF₁ × PF₂ = a²[br][br]Construction Steps[br]1. Place two foci F₁ and F₂ on the x-axis (e.g., F₁ = (−1,0) and F₂ = (1,0)).[br]2. Define the midpoint O of the foci, which serves as the origin of symmetry.[br]3. Let d = distance(F₁, F₂). This distance determines the scale of the lemniscate.[br]4. Define the parameter a = d / √2. The constant a is tied to the foci separation.[br]5. Construct the lemniscate in Cartesian form: (x² + y²)² = (d²/2)(x² − y²).[br]6. Place a point P on the curve (Point[Lemniscate] in GeoGebra).[br]7. Measure PF₁ × PF₂ and compare it with a² to verify the definition.[br]8. Add an optional reference circle with center O and radius a√2 for symmetry visualization.[br][br]Drag Test[br]The drag test confirms that the lemniscate maintains its defining property under dynamic manipulation:[br]• Dragging F₁ or F₂: The curve automatically updates in scale and orientation.[br]• Moving point P along the lemniscate: The product PF₁ × PF₂ remains equal to a².[br]• Observing the circle centered at O with radius a√2:[br]Provides a geometric boundary reference. Through dragging, students can visually verify that[br]the locus definition holds true, strengthening the connection between algebraic equations and geometric meaning.