Functions
AI Assistant

The Ellipse

Foci, eccentricity, and the string property — one slider at a time

An ellipse is like a circle that's been stretched in one direction. Its equation is \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, where a is the horizontal radius and b is the vertical radius. When a = b, you get a perfect circle.

Every ellipse has two special interior points called foci. The magic property: if you pick any point on the ellipse, the sum of its distances to the two foci is always the same. This is why you can draw an ellipse with two pins and a loop of string — the string keeps the total distance constant.

In this lesson, you'll explore how sliders for a and b reshape the ellipse, locate the foci, verify the string property, and discover eccentricity — the number that tells you how "squished" the ellipse is. Every planet's orbit is an ellipse!

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FAQ

What is an ellipse?
An ellipse is a stretched circle, defined by the equation \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. The value a is the horizontal semi-axis (half-width) and b is the vertical semi-axis (half-height). When a = b, the ellipse becomes a circle.
What are the foci of an ellipse?
The foci (singular: focus) are two special points inside the ellipse. For a horizontal ellipse with a > b, the foci are at (\pm c, 0) where c = \sqrt{a^2 - b^2}. The defining property: the sum of distances from any point on the ellipse to the two foci equals 2a (constant).
What is eccentricity?
Eccentricity e = \frac{c}{a} measures how elongated the ellipse is. When e = 0, it's a perfect circle. As e approaches 1, the ellipse becomes very elongated. Earth's orbit has e ≈ 0.017 (nearly circular), while Halley's comet has e ≈ 0.967 (very elongated).
What is the string property of an ellipse?
The string property says: for any point P on the ellipse, the sum of distances from P to the two foci is always 2a (twice the semi-major axis). This means you can draw an ellipse by pinning a string of length 2a at both foci, pulling it taut with a pencil, and tracing — the pencil draws a perfect ellipse.

Related Topics

The Unit CircleQuadratic EquationsPythagorean Theorem