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
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
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.
- 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'?
- 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?
- Drag the top vertex. How do your changes affect the
shape of the hyperbola? What happens to the value of the difference of
- What happens to the shape of the hyperbola when one
vertex is very close to a focus? Far away from a focus?
- How can you find the equation of a hyperbola using
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
- 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.
- 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
- 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?
- 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?
- Find the equations of the asymptotes of the unit
- Press Show Point on Hyperbola. What happens
to the coordinates of P when the point is dragged away from