Conic sections by eccentricity
MathematicsVectors & Coordinate GeometryAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Vectors & Coordinate Geometry
With the same focus and directrix, only the eccentricity e changes: e < 1 gives an ellipse, e = 1 a parabola, e > 1 a hyperbola. The simulation draws the conic, marks the focus and axis, and lets you check the property MF = e·d(M, Δ) at a point M you can drag along the curve.
Lesson: Conic sections
What it shows
Ellipses, parabolas and hyperbolas belong to one family. Each is the set of points M whose distance to a fixed focus F is e times the distance to a fixed line, the directrix Δ: MF = e·d(M, Δ). In polar form about the focus, r = p/(1 + e·cos θ), where p is the semi-latus rectum. For e < 1 the curve closes into an ellipse that becomes rounder as e approaches 0; at e = 1 it opens into a parabola; for e > 1 it splits into the two branches of a hyperbola. Changing p only rescales the curve without changing its shape.
How to use
Start with the three preset buttons, e = 0.6 (ellipse), e = 1 (parabola) and e = 1.6 (hyperbola), then drag Eccentricity e slowly through 1 and watch the ellipse stretch, open and split into two branches. Keep the check switched on and drag point M along the curve: the readings MF and e·d stay equal. Finally vary Semi-latus rectum p to show that it changes the size but not the shape.
Parameters you can change
- Eccentricity e 0.1–2.5
- Semi-latus rectum p 1–8
- Show check: MF = e·d(M, Δ)
Questions to explore
- With e = 1, which curve does the equality MF = d(M, Δ) define, and why does it not close?
- What shape does the ellipse approach as e gets closer to 0?
- Why does changing p alter the size of the conic but not its shape?
