ASSIGNMENT 6

Problem # 9

A parabola is defined as the set of all points eqidistant from a line, called the directrix, and a fixed point, called the focus. We are assuming that the focus and directrix are not on the same line. The goal of this lesson is to construct a parabola using only the focus and a line-segment (the directrix).

I have chosen to include a list of instructions on how to use GSP to construct such a parabola.
STEP 1: Draw a line and designate it as the directrix (line j).
STEP 2: Construct a point that is not located on the directrix and designate it as the focus.
STEP 3: Select the directrix and construct a point on the directrix (C).
STEP 4: With the new point selected as well as the directrix, construct a perpendicular line through the point (line k).
STEP 5: Construct a segment from the focus to the arbitrarty point constructed in step 2.
STEP 6: Construct the midpoint of the segment constructed in step 5 and proceed to construct a perpendicular line through that point (E).
STEP 7: Now, the line that is perpendicular the directrix and the perpendicular bisector of the
segment constructed in step 5 will intersect (F).

The final construction should resemble the following picture.

Once these preliminary constructions are completed, it is then necessary to animate point C along line j. One way to show the construction of the parabola is to trace point F. By doing so, the set of all points eqidistant from the dirrctrix and the focus is being constructed.

click here for parabola

An alternative approach would be to perform the same animation, but trace line k such that the envelope of lines formed by this animation will construct the parabola. Indeed, these lines are tangent lines to the parabola at any given point.

click here for parabola

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