Ellipses

Lesson notes

Overview

A stretched circle: the sum of distances to two foci stays constant. Planets travel on ellipses. This topic is rated beginner and is number 31 of 245 in Mathematics. Core ideas and first intuitions. No prior knowledge needed.

What you are seeing

The panel above shows a path traced by a parametric curve as its parameters change. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

An ellipse is a circle stretched in one direction. Its equation x²/a² + y²/b² = 1 has semi-axes a and b. Equivalently, it is the set of points whose distances to two fixed foci add up to a constant. The eccentricity measures how stretched it is, from 0 for a circle toward 1. Kepler found that planets move on ellipses with the Sun at one focus, and a sound or light source at one focus is reflected to the other.

The key idea, step by step

1. A stretched circle: the sum of distances to two foci stays constant. 2. Planets travel on ellipses.

Reading the relationship

The expression "x²/a² + y²/b² = 1" is a compact summary of the idea. It uses the symbols x, a, y, b; name what each one stands for before you start. Notice that it contains a squared term, so doubling that quantity changes the result by a factor of four, and a ratio, so increasing the denominator makes the result smaller. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.

Try this

Start by only moving one slider at a time. Say aloud what you expect, then check.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Circles”, the topic before it in Mathematics. Once you are comfortable, continue with “Parabolas as Conics”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.