Activity 2-5: Conics

Take a point A, and a line not through A. Another point B moves so that it is always the same distance from A as it is from the line.

Task: what will the locus of B be? Try to sketch this out. This looks very much like a parabola...

We can confirm this with coordinate geometry: This is of the form y = ax 2 + bx + c, and so is a parabola.

Suppose now we change our starting situation, and say that AB is e times the distance BC, where e is a number greater than 0. What is the locus of B now? We can use a Geogebra file to help us. Geogebra file

We can see the point A, and the starting values for e and q (B is the point (p, q) here). What happens as you vary q? The point B traces out a parabola, as we expect. (Point C traces out the left-hand part of the curve.)

Now we can reduce the value of e to 0.9. What do we expect now? This time the point B traces an ellipse. What would happen if we increased e to 1.1? The point B traces a graph in two parts, called a hyperbola.

Can we get another other curves by changing e? The ellipse gets closer and closer to being a circle. What happens as e gets closer and closer to 0? What happens as e gets larger and larger? The curve gets closer and closer to being a pair of straight lines.

So to summarise: This number e is called the eccentricity of the curve. e = 0 – a circle. 0 < e < 1 – an ellipse. e = 1 – a parabola 1 < e <  – a hyperbola. e =  – a pair of straight lines.

Now imagine a double cone, like this: If we allow ourselves one plane cut here, what curves can we make? Clearly this will give us a circle.

This gives you a perfect ellipse… A parabola… A hyperbola… and a pair of straight lines. Exactly the same collection of curves that we had with the point-line scenario.

This collection of curves is called ‘the conics’ (for obvious reasons). They were well-known to the Greeks – Appollonius (brilliantly) wrote an entire book devoted to the conics. It was he who gave the curves the names we use today.

Task: put the following curve into Autograph and vary the constants. How many different curves can you make? ax 2 + bxy + cy 2 + ux + vy + w = 0 Exactly the conics and none others!

Notice that we have arrived at three different ways to characterise these curves: 1. Through the point-line scenario, and the idea of eccentricity 2. Through looking at the curves we can generate with a plane cut through a double cone 3. Through considering the Cartesian curves given by all equations of second degree in x and y. Are there any other ways to define the conics?

With thanks to: Wikipedia, for helpful words and images. Carom is written by Jonny Griffiths,