The Ellipse

Two foci and a string

An interactive lesson with 8 steps, about 7 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• An ellipse: the points where $d_1 + d_2 = 2a$ (constant).
• $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, with foci at $\pm c$, where $c^2 = a^2 - b^2$, and $e = c/a$.
• Rays from one focus reflect to the other. Planets orbit with the Sun at a focus.

Steps

1. The gardener’s ellipse
2. The equation
3. Drawing an ellipse with a stick
4. Find the foci
5. Orbits
6. Whispering galleries
7. A whisper at the focus
8. Ramanujan’s perimeter