Activity 2-5: Conics

www.carom-maths.co.uk

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 = ax2 + 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 parabola1 < 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 perfectellipse…

A parabola… A hyperbola…

and apair 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?

ax2 + bxy + cy2 + ux + vy + w = 0

Exactly the conicsand 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, mail@jonny-griffiths.net