A parabola is the set of points equidistant from a line, called the directrix, and a fixed point, called the focus. Assume the focus is not on the line. Construct a parabola given a fixed point for the focus and a line (segment) for the directrix.
a. Use an Action Button to generate the parabola from an animation and trace of a constructed point.
b. Repeat part a with a trace of the tangent line at the constructed point.
c. Use the locus command to generate the parabola from a constructed point of the tangent line at that point.
First, how to use GSP to construct a parabola.
1. Construct a line, a point on that line, and a perpendicular line through that point.
2. Construct a point off the line. Construct a line segment from the point off the line to the point on the line with the perpendicular line through it.
3. Find the midpoint of that segment and construct a perpendicular line through that point.
4. The intersection of the two perpendicular lines as the first point is animated along the first line (the directrix) will form the parabola.
Part a asks for a trace of the intersection of a point. Part b asks for a trace of the line.
Part c, construction using the locus command, is can be created by selecting the point of intersection of the two perpendicular lines and the point along the directrix. Then, under the construct menu there is the locus command option. Creating the parabola using the locus comman is really neat because any changes that you make to the focus, for example, are immediately evident. The locus of the parabola changes immediately and you do not have to erase the traces and reanimate the point.
Here is the GSP file. There are three pages to this document, which correlate to the three parts of this prompt.
But can we continue this exploration and look at the set of points equidistant from something other than a line...?
What about a circle?
The set of points equidistant from a circle is an ellipse. Click here for the GSP file to explore. As the point that is on the circumference of the circle in the image above moves around the circle, the red ellipse is traced out by the yellow dot, the point that is equidistant from the point on the circle and point A. The construction of this figure is remarkably similar to the parabola construction above. Note here that point A is inside the circle.
What would happen is A is outside the circle?
Click here for the GSP file to explore.
We find that the set of equidistant points from a point on the circle and a point outside the circle is a hyperbola.
By definition of ellipses and hyperbolas, these conclusions make sense. The loci of the curves that are traced out represent the constant, fixed distances from the point A and the point on the circle.
