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Constructing a Parabola

On this web page you can explore the geometric construction of a parabola. In Lesson 9.3 of Discovering Advanced Algebra: An Investigative Approach, you learned the locus definition of a parabola.

Definition of a Parabola

A parabola is a locus of points P in a plane, whose distance from a fixed point, F, is the same as the distance from a fixed line, <em>l</em>. The fixed point, F, is called the focus. The line, <em>l</em>, is called the directrix.

The sketches on this page use this definition to construct a parabola. The construction is the same as the one suggested in Question 3 of the Exploration Constructing the Conic Sections.

Sketch

This sketch shows a focus, F, and a directrix. Point B is any point on the directrix. Point P is the intersection of the perpendicular bisector of FB and the line perpendicular to the directrix through B. Drag point B to trace the intersection P. Drag point F to reposition the focus, and drag the orange points to reposition the directrix. Click X to erase the traces.

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Investigate

  1. Slowly drag point B back and forth along the directrix. What shape is created by the trace of the intersection P?
  2. What is true about segment FP and segment BP? How does the construction create this relationship? (Hint: What type of triangle is FPB?)
  3. Why is the trace of the intersection a parabola?
  4. Try moving F and/or the directrix, erase the traces, then slowly drag B back and forth. Do you still get a parabola? In what ways has it changed?

Sketch

This sketch shows the locus of intersection P from the previous sketch. Drag point F to reposition the focus. (Note: For technical reasons, you can't reposition the directrix in this sketch. Also, for some positions of the focus, you may get erroneous vertical lines.)

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Investigate

  1. Experiment by dragging the focus farther from and closer to the directrix. How does the parabola change?
  2. How can you make the parabola open downward instead of upward?
  3. What would need to change in order for the parabola to open to the left or to the right?
  4. What happens when the focus is on the directrix?
  5. How can you make the parabola narrower? Wider? Explain this behavior based on the locus definition of a parabola.
  6. If you could drag the focus infinitely far away from the directrix, what do you think would happen to the parabola?

Go to the ellipse page or the hyperbola page to explore the geometric constructions of other conic sections.