This is an investigation of the construction of a parabola. I have started with a line, the directrix, a point on the line, and a point in the plane not on the directrix, the focus. The parabola consists of the set of points equidistant from the focus and directrix. Click on the animate button to check it out.
The same parabola can be created by tracing the lines tangent to every point in the set of points creating the parabola.
Using the locus command under the construct menu in Geometer's Sketchpad, I have created the set of all points of a parabola. Move the focus around and see the infinite number of possible parabolas. Don't forget to move it under the directrix!
Below I have used the locus command again to create not only a parabola, but also an ellipse, and hyperbola. As you can see, I have used the Greek construction rules to create these conics. Note for a parabola, the distance of PF (point to focus) equals the distance PD (point to directrix). You've probably noticed the circles in this construction. The original circle used to construct the parabola is controlled by the slider in the top left corner. I constructed the ellipse using a circle 1/2 the size of the circle used to construct the parabola. I constructed the hyperbola using a circle 2 times the size of the circle used to construct the parabola. 1/2 and 2 are the eccentricities of these equations.
PF=PD ==> gives a parabola
PF=1/2PD ==> gives an ellipse
PF=2PD ==> gives a hyperbola