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Activity 2-6: 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 quadratic curve ...
We can confirm this with coordinate geometry:
This is of the form y = ax2 + bx + c, and so is a quadratic curve.
We call this curve a parabola.
A curve is a parabola with a vertical axis  it is a quadratic curve.
Suppose now we change our starting situation,
and say that AB is e times the distance BC,
where e is a non-negative number called the eccentricity.
What is the locus of B now?
We can use the Geogebra file below to help us
(the eccentricity is called a here – Geogebra reads e as the famous number e).
Geogebra Conics File
http://www.s253053503.websitehome.co.uk/
carom/carom-files/carom-2-6.ggb
When a is 1, the point B
traces out a parabola,
as we expect.
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
a to 1.1?
The point B traces
a graph in two
parts, called a
hyperbola.
Can we get another other curves by changing e?
What happens as a gets larger and larger?
The curve gets
closer and
closer to being
a pair of
straight lines.
What happens as a
gets closer and closer
to 0?
The ellipse gets closer
and closer to being
a circle.
So to summarise:
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, with its axis vertical:
If we allow ourselves
one plane cut here,
what curves can we make?
How about a horizontal cut?
Clearly this will give us a circle.
This gives you
a perfect
ellipse…
A parabola…
A hyperbola…
Exactly the same collection of curves
that we had with the point-line scenario.
and a
pair of
straight
lines.
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 conics
and none others!
Notice that we have arrived at three different ways
to characterise this set of 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, [email protected]