In this lesson students will become familiar with the equations and graphs of hyperbolas.† The definition of a hyperbola will be learned both algebraically and using the distance relationship.† Students will learn how to construct a hyperbola using Geometerís Sketchpad and how to prove that this construction is a hyperbola.† Applications of hyperbolas will be explored.
Definition:† A hyperbola is the set of points P(x, y) in a plane such that the absolute value of the difference between the distances from P to two fixed points in the plane, F1 and F2, called the foci, is a constant.
Some other important information includes the following:
∑ The center of the hyperbola is located at (h, k).
∑ The vertices are located on the transverse axis.† The length of the transverse axis is 2a.
∑ The co vertices are located on the conjugate axis.† The length of the conjugate axis is 2b.
∑ The foci are located on the transverse axis a distance of c from the center.
∑ The equations for the asymptotes of the hyperbola are for a horizontal transverse axis hyperbola, and for a vertical transverse axis hyperbola.
∑ The relationship between a, b, and c is a2+b2=c2.
Standard equations for the hyperbola and the graph of each are shown below.
Note that when the center of the hyperbola is translated to a point (h, k) so that the center is other than the origin, the equation will become
†for figure 12, and †for figure 13.
Try constructing a hyperbola using GSP.† †Use the instructions provided to complete this construction by clicking here.† After your construction is complete, prove that the construction is a hyperbola.† Show your proof using two different methods, geometric and algebraic.† Click here to explore the construction of a hyperbola on GSP.† The hyperbola that has been constructed is not necessarily oriented to the standard coordinate system.
Practice problem 1:† An explosion is heard by two law enforcement officers who are 1000 meters apart.† One officer heard the explosion 1.5 seconds after the other officer.† The speed of sound in air (at 20 C) is approximately 340 meters per second.† Write an equation for the possible locations of the explosion, relative to the two law enforcement officers.
Practice problem 2:† Translate the hyperbola defined by the equation 9x2-4y2+54x+8y+41=0 up 2 units and to the left 6 units.† Write the standard equation of the resulting hyperbola.
Practice problem 3:† A telescope may have a hyperbolic mirror with the property that a ray of light directed at one focus is reflected to the other focus.†† If the center is located at the origin, a focus has coordinates (5, 0), a vertex has coordinates (3, 0), and one end of the mirror is attached to the telescope at the point (5, 16/3), find the equation that defines the mirror.† Draw a diagram.