Conic Sections Explorer
Interactive conic sections explorer with adjustable semi-major/semi-minor axes. Visualizes foci (red), vertices (green), directrix (amber), and asymptotes. Shows standard form equation, eccentricity, and focal distance for Circle, Ellipse, Parabola, and Hyperbola.
CONCEPT: PLANAR SLICES
Conic sections are the curves formed by the intersection of a plane and a double-napped cone. Depending on the angle of the plane, you get a circle, ellipse, parabola, or hyperbola. In Precalculus, we define these curves algebraically and through their "locus" properties (distance from foci).
MECHANISM: FOCI AND ECCENTRICITY
Every conic can be defined by its eccentricity (). - **Circle ()**: Constant distance from one center. - **Ellipse ()**: Constant sum of distances from two foci. - **Parabola ()**: Equidistant from a focus and a directrix. - **Hyperbola ()**: Constant difference of distances from two foci.
HOW TO USE THIS VISUALIZATION
1. **Switch Conic Type**: Toggle between the four conic shapes. 2. **Adjust Parameters**: Change the values of and to see how the foci move and how the "flatness" (eccentricity) changes. 3. **View the Locus**: Watch the animated lines that show the distance sum/difference property as a point moves along the curve.
CORE FORMULAS
AP EXAM CONNECTION
Unit: Unit 4: Analytic Geometry (Topic 4.1-4.3)
Learning Objective: LO 4.1.A
COMMON MISCONCEPTIONS
- Confusing the major and minor axes in an ellipse.
- Thinking a parabola has two foci like an ellipse.
- Forgetting that for ellipses but for hyperbolas.
KEY TAKEAWAYS
- The eccentricity determines the shape of the conic.
- Foci are the "anchors" that define the curvature.
- Parabolas are the only conics with a directrix and exactly one focus.
PRACTICE QUESTIONS
Q1 (CONCEPTUAL): If , what type of conic section is it?
Show Answer & Explanation
Answer: Ellipse
Explanation: An ellipse has an eccentricity between 0 and 1.
