4.
The ellipse
4.1
Exploring the ellipse
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An ellipse is defined as
the locus of points so that the sum of its distances to two fixed
points (called the foci) is constant.
This is often
called the tack-and-string definition, because it can be used to
easily draw an ellipse: fix a piece if string with two tacks at
F1 and F2, stretch the string tight with a
pencil at P, and then move the pencil while keeping the string tight.
The resulting curve will be an ellipse, because F1P +
F2P equals the length of the string and is therefore
constant. Drag F1 and F2 and check ...
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Open the Exploring the ellipse applet: 
Click on the construction
buttons to see how an ellipse can be constructed. Animate point S …
We will visually and
numerically investigate the definition, and then prove that the construction
implies that F1P + F2P is constant, and therefore
that the locus is an ellipse by definition.
Drag F2
to different positions inside the circle – the ellipse is not defined
when F2 is on or outside the circle – why do you think
that is? Do not now move F1.
How does the shape of the ellipse change? How will you describe these
different shapes in words?
When is the
ellipse a circle? What can you deduce from this?
Note that F1P
+ F2P remains constant for any position of F2 inside
the circle. Can you explain why?
Click “Show
Q”. Drag point Q. The measurements are not very accurate, but convince
yourself that:
- for Q inside the
ellipse, F1Q + F2Q < F1P + F2P
- for Q outside
side the ellipse, F1Q + F2Q > F1P
+ F2P
- for Q on the ellipse,
F1Q + F2Q = F1P + F2P
Now drag F1.
Note that F1P + F2P changes. Why is that?
Now prove, using simple Synthetic Geometry, that F1P
+ F2P = F1S, the radius of the circle, and therefore
it is constant and therefore the locus is an ellipse by its definition.
Click here for some
discussion: 
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