The Hyperbola

 Definition:  A hyperbola is the set of points such that the difference of the distances from each point to two fixed points (the foci) is constant.

Recall that for an ellipse the sum of the distances from a point on the curve to the two foci remains constant, but note for a hyperbola, it is the difference of the two distances that remains constant.

To make a Sketchpad model for drawing a hyperbola, begin by opening the following Two-Pins –and-a-String Construction for an ellipse.  Review the steps of construction and the reason this construction works.  It will not take much to change this model so that it will draw a hyperbola.

Open the following construction to alter the ellipse sketch slightly in order to draw a hyperbola.

The following information should be given to the students through direct teaching

A hyperbola is the set of all points, P, in a plane such that the difference from

P to two fixed points, called foci, is a constant.

The standard form of the hyperbola with center (h,k) is

or

They hyperbola is similar in some of its characteristics to the ellipse, but very different in others.

The foci for the hyperbola are defined by the relationship

The latus rectum relationship is the same for both the hyperbola and the ellipse.

For each graph, list the vertices, the foci, the latus rectum distance and the center.  As you graph the following hyperbolas, watch for similarities and differences between this curve

and the ellipse listed next to it.

 Hyperbola Ellipse Answer Click to Compare Click to Compare Click to Compare

Answer the question and do the activities that follow:

1.               What do you notice about the relationship between the size of a and b on the hyperbola that is different on the ellipse?

2.               What seems to determine the direction of the hyperbola?

3.               What determines the direction of the ellipse?

4.               Compare and contrast as many characteristics of the two curves as you can.

5.               Write the equations for each of the following hyperbolas and check them on the computer.

a.    hyperbola (h,k) =(2,3) a=4, b=9 lies in the x-direction

b.    hyperbola (h,k) =(-1,-2) a=9, b=4 lies in the y-direction

c.     hyperbola (h,k) =(4,5) a=4, c=16 lies in the y-direction