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, F_{1} and F_{2}
are the foci. You can drag the focus F_{1} 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.
Investigate
 What happens to the difference of the distances from P
to F_{1} and to F_{2} when you move
point P? How are the distances d_{1} and d_{2}
related? What about when you move point P'?
 Drag the focus F_{1}. 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
the distances?
 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
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 righthand focus and vertex and observe how the
hyperbola changes.
Investigate
 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 upperright corner of the asymptotic rectangle
closer to the origin than the focus, farther away, or the same
distance? Defend your opinion.
 Call the xcoordinate of the hyperbola's
right vertex b and the xcoordinate of its right
focus c. What are the coordinates of the upperrighthand
vertex of the asymptotic rectangle in terms of b and c?
What are the coordinates of the other vertices of the asymptotic
rectangle?
 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
hyperbola.
 Press Show Point on Hyperbola. What happens
to the coordinates of P when the point is dragged away from
the vertex?
