Generate the Parabola
from the Definition
By
Ronnachai
Panapoi
In this topic, I will present how to
generate the locus of the parabola by using the definition to generate the locus
directly. By this way, I think the students will find some relations in
geometry from the construction. Also, I believe that the students will see the
result from the definition and have more understanding in the definition of the
parabola.
To
generate the locus, I present by using the Geometer’s Sketchpad 4.06 as a tool.
Definition of Parabola
“A parabola is the set of points
equidistant from a line, called the directrix, and a
fixed point, called the focus.”
From
the definition of the parabola, we have the idea that to generate the locus we
firstly must have a line called the directix and a
fixed point called the focus.
So,
construct the line (The Directrix) and the fixed point(The Focus) not lying on the line.
Click
HERE to follow the construction.
From
the definition, the locus have the property that the distance between it and
the focus equals to the distance between it and the derectrix.
We
have to think of the theorem state that the point on the perpendicular bisector
of a segment is equidistant form the endpoints.
So,
we construct the segment between the focus and the free point on the directrix. Then, construct the perpendicular bisector.
Click
HERE to see the construction.
Then,
we find the point which will show the locus of the parabola by construct the
perpendicular line of the directrix at the free point
on it. After that, we draw a segment from the intersection point to the focus.
We
will see that the intersection point is equidistant from the focus and the directrix that corresponds to the definition of the
parabola.
Finally,
after the free point on the directrix moves, the
intersection point will give us the locus of the parabola.
Click
HERE to see the locus occurring form the movement of
the free point lying on the directrix.
With
the same way, we can construct the parabola with the trace of the tangent line
at the constructed point.
Click
HERE to see.