By: Melissa Wilson
In this assignment we will explore how to construct a parabola and use GSP to do so. First, we will look at the definition of a parabola. A parabola is the set of all points that are equidistant from a single point (called the focus) and a line (called the directrix). We will look at the case where the focus is not on the directrix.
Now, we will need to pick a random point on the directrix (line AB), which we will call D. We will need the perpendicular line from AB through D as well. The set of points for our parabola will always fall on this perpendicular line as we move D along AB.
Next we will need the midpoint from the focus to D. We will do this by constructing the segment from D to the focus and finding the midpoint, labeled M below.
The last step is constructing the perpendicular line to this segment through M. Now we label intersection of the two perpendicular lines as P. This is the point on our parabola.
Let's check to make sure the construction achieved what we wanted it to. P should be the same distance from D (on AB, the directrix) as it is from the focus. If we treat the line segment from D to the focus as the base of a triangle, we can see that P lies on the altitude of that triangle (since it is on the perpendicular from the base). This triangle will be isosceles since the altitude goes through the midpoint of the base. Hence, the distance from D to P is the same as the focus to P.
Here is the GSP file used, with an animate button. If you trace P as D is moved along AB, then you get the resulting picture. This is our parabola.
Another exploration includes looking at the line tangent to the parabola. We will do this by tracing the perpendicular line we constructed through M. By construction, this line will be perpendicular to the parabola at each point P. You can access this animation in this GSP file. Below is an image of the result. The blue lines below the parabola are the tangent line traces.
GSP can make this trace a lot easier. After finding P, you can make the parabola by using the "Locus" command under the construct menu. You will need to select D (movable point) and P (the point that depends upon D). Below is what you get when you use this feature.
You can also do this with the perpendicular line and point D:
