activity 2-5: conics . take a point a, and a line not through a. another point b moves so that it is...
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Activity 2-5: Conics
www.carom-maths.co.uk
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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.
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Task: what will the locus of B be?Try to sketch this out.
This looks very much like a parabola...
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We can confirm this with coordinate geometry:
This is of the form y = ax2 + bx + c, and so is a parabola.
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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
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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.)
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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.
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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.
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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.
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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.
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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.
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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.
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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!
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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?
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With thanks to:Wikipedia, for helpful words and images.
Carom is written by Jonny Griffiths, [email protected]