4.5
The reflective property of the ellipse
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The reflective
property of the ellipse is one of its most useful properties.
Analyse
the equal angles in the applet …
How would you describe the reflective property?
How does this compare with the reflection of light in a mirror,
where we know that the angle of incidence is equal to the
angle of reflection?
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In
the above applet, the pink line is a tangent to the ellipse.
But how do we prove that it is a tangent?
Here is the beginning
of a proof: Assume it is not tangent,
but that the line cuts the ellipse in another point Q. Draw the
lines as shown.
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So
if Q is a point on the ellipse, it must satisfy the definition of
the ellipse, i.e.
F1Q + F2Q = F1P + F2P.
In the applet, drag point Q – when is
F1Q + F2Q = F1P + F2P?
You should notice
from the measurements that
F1Q + F2Q >
F1P + F2P, except when Q falls on P.
Prove
that F1Q + F2Q > F1P + F2P.
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Click here to see
a proof: 
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The tangent can be
used to construct an ellipse as an envelope of tangents, just like we constructed the parabola as an envelope.
In the applet, click on “Show envelope” ...
Drag F1 and F2
and reflect on what you see!
Beautiful is it not!?
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