Lemniscate of Bernoulli Quiz
Lemniscate of Bernoulli: Construction, Drag Test, and Observations
Description of the Construction
The Lemniscate of Bernoulli is a degree-four curve shaped like a sideways figure-eight (∞). It is defined as the locus of all points P such that the product of the distances to two fixed points (the foci)
F₁ and F₂ is constant:
PF₁ × PF₂ = c²
Construction Overview:
1. Two foci F₁ and F₂ are placed on the x-axis.
2. The midpoint O of the foci defines the center of symmetry of the lemniscate.
3. The distance between the foci is 2c, and the constant of the locus is c².
4. The lemniscate is drawn using the Cartesian equation:
(x² + y²)² = 2c²(x² − y²)
5. A test point P is placed on the curve, and GeoGebra dynamically computes both PF₁ × PF₂ and c² for comparison.
Drag Test
• Drag the foci F₁ or F₂ along the x-axis. The curve resizes but keeps its shape (the product property still holds).
• Move the point P around the lemniscate. The computed product PF₁ × PF₂ stays constant and equal to c².
• The red circle centered at O with radius √2c provides a visual reference for the outer boundary.
This drag test allows you to confirm dynamically that the lemniscate maintains its geometric relationship for all positions of P and for any adjustment of the foci.
Multiple-Choice Questions
1. What is the geometric definition of the Lemniscate of Bernoulli?
2. If the distance between the foci is 2c, what constant defines the lemniscate?
3. During the drag test, when the foci are moved farther apart, what happens to the lemniscate?
4. What does the drag of point P along the curve demonstrate?
5. Which statement about the curve’s symmetry is true?