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Properties of Hyperbolas

On this web page you will investigate the properties of hyperbolas. This exploration will help you understand the concepts in Lesson 9.4 of Discovering Advanced Algebra: An Investigative Approach.

A hyperbola is a locus of points in a plane, the difference of whose distances from two fixed points is a constant. The two fixed points are called the foci of the hyperbola.

Sketch

In the sketch below, F1 and F2 are the foci. You can drag the focus F1 and the upper vertex. The hyperbola that they determine will be shown. You can drag points P and P' on the hyperbola and see the difference of the distances from those points to the two foci.

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Investigate

  1. What happens to the difference of the distances from P to F1 and to F2 when you move point P? How are the distances d1 and d2 related? What about when you move point P'?
  2. Drag the focus F1. How do your changes affect the shape of the hyperbola? What happens to the value of the difference of the distances?
  3. Drag the top vertex. How do your changes affect the shape of the hyperbola? What happens to the value of the difference of the distances?
  4. What happens to the shape of the hyperbola when one vertex is very close to a focus? Far away from a focus?
  5. How can you find the equation of a hyperbola using its definition?

Sketch

With this sketch, you can investigate the properties of the asymptotes of a hyperbola. An asymptote is a line that one of the arms of the hyperbola gets closer to. In the sketch below, you can drag the right-hand focus and vertex and observe how the hyperbola changes.

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Investigate

  1. Press Show Asymptote Rectangle. The rectangle that appears passes through the two vertices of the hyperbola. It is also inscribed in a circle centered at the origin and passing through the two foci. Is the upper-right corner of the asymptotic rectangle closer to the origin than the focus, farther away, or the same distance? Defend your opinion.
  2. Call the x-coordinate of the hyperbola's right vertex b and the x-coordinate of its right focus c. What are the coordinates of the upper-right-hand vertex of the asymptotic rectangle in terms of b and c? What are the coordinates of the other vertices of the asymptotic rectangle?
  3. Press Show Asymptotes to see the two diagonal lines of this rectangle. These lines are the asymptotes of the hyperbola. What are equations of the asymptotes?
  4. Press Unit Hyperbola to see a hyperbola whose vertices are at (???1, 0) and (1, 0) and whose foci are at and . What is the equation of this hyperbola? Why do you think it is called the unit hyperbola?
  5. Find the equations of the asymptotes of the unit hyperbola.
  6. Press Show Point on Hyperbola. What happens to the coordinates of P when the point is dragged away from the vertex?