The Hyperbola

 

 

Introduction

  

 

Objectives

  1. Sketching the Graph of a Hyperbola
  2. Determining the Equation of a Hyperbola in Standard Form
  3. Translated hyperbolas
  4. Solving Applied Problems Involving Ellipses
  5. Classifying Equations of Conic Sections

The activities in this lesson are not graded. However, I strongly advised you to complete them to prepare for the quiz ANGEL.

  

 

Objective 1: Sketching the Graph of a Hyperbola

When a plane intersects two right circular cones at the same time, the conic section formed is a hyperbola.  

 

conic-hyperbola2.jpg

 Hyperbola

 

The Geometric Definition of the Hyperbola

 

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points is a positive constant.   The two fixed points, F1 and F2 , are called the foci.The midpoint of the line segment joining the foci is called the center of the hypebola.

 

 

Notice that the geometric definition above is very similar to the geometric definition of the ellipse.   Recall that for a point to lie on the graph of an ellipse, the sum of the distances from the point to the two foci is constant.   For a point to lie on the graph of a hyperbola, the difference between the distances from the point to the two foci is constant.   Because subtraction is not commutative, we will consider the absolute va lue of the difference in the distances between a point on the hyperbola and the foci to ensure that the constant is positive.   Thus for any two points P and Q  that lie on the graph of a hyperbola

 

 

For any points P and Q that lie on the graph of a hyperbola,

 

 

The graph of a hyperbola has two branches.   Each branch looks somewhat like a parabola but are certainly not parabolas because the branches do not satisfy the geometric definition of the parabola.   Every hyperbola has a center, two vertices and two foci.  

 

The vertices are located at the endpoints of an invisible line segment called the focal axis or transverse axis.   The transverse axis is either parallel to the x-axis (horizontal transverse axis) or it is parallel to the y-axis (vertical transverse axis).

 

The center of a hyperbola is located midway between the two vertices (or two foci).   The hyperbola has another invisible line segment called the conjugate axis which passes through the center and lies perpendicular to the transverse axis.

 

Each branch of the hyperbola approaches (but never intersects) a pair of lines called asymptotes. These lines intersect at the center of the hyperbola and have the property that as a point P moves along the hyperbola away from the center, the distance between P and one of the asymptotes approaches zero.

A reference rectangle is typically used as a guide to help sketch the asymptotes.   The reference rectangle is an imaginary rectangle whose midpoints of each side are the vertices of the hyperbola or the endpoints of the conjugate axis.   The asymptotes pass diagonally through opposite corners of the reference rectangle.

 

SUMMARY PARTS OF A HYPERBOLA

Focal axis or Transverse axis:

                      Parallel to the x-axis if the hyperbola is horizontal

                      Parallel to the y-axis if the hyperbola is vertical

 

Conjugate axis:   Passes through the center and is perpendicular to the transverse axis.

 

Vertices:   Located at the endpoints of the transverse axis

 

Center:   Midpoint of the segment joining the two vertices and the line segment joining the two foci.

 

Reference Rectangle :   Rectangle whose midpoints of each side are the vertices of the hyperbola or the endpoints of the conjugate axis.

 

Asymptotes:   Pass diagonally through opposite corners of the reference rectangle.   Each branch of the hyperbola approaches (but never intersects) the asymptotes

Description: E:\artSection6.3_Trigsted\figure22a.jpg   (a)   Hyperbola with a Horizontal Transverse axis

 

 

                                                               

Description: E:\artSection6.3_Trigsted\figure22b.jpg

 

  (b)   Hyperbola with a Vertical Transverse Axis

                             

Objective 2: Determine the Equation of a Hyperbola in Standard Form

It is traditional in the study of hyperbolas to denote the distance between the vertices by 2a, the distance between the foci by 2c (see the fiqure below) and to define the quantity b as

 

hyperbola1.png

 

To derive the equation of a hyperbola, consider a hyperbola with a horizontal transverse axis and following the below Geometric Properties of a Hyperbola Activity

 

By the geometric definition of a hyperbola we know that for any point P(x,y) that lies on the hyperbola, the difference of the distances from P to the two foci is a constant.   From the Geometric Properties of a Hyperbola Activity, it shown that this constant is equal to 2a, the length of the transverse axis.  

 

Hyperbola has the follow properties

  1. Given the foci of a hyperbola F1 and F2 and any point P that lies on the graph of the hyperbola, the difference of the distances between P and the foci is equal to 2a .   In other words, .   The constant 2a represents the length of the transverse axis.
  2. We define the cooridinate as F1(-c,0) and F2(c,0)The equation of a hyperbola is .......

 

Since , we use the distance formula and abtain

 

then square both sides and simplifying many more steps (I won't go through the detail) yields

 

 

Standard Equation of a Hyperbola with the Center at the Orgin:

The equation of a hyperbola is simplest if the coordinate axes are posititioned so that the center of the hyperbola is at the orgin and the foci are on the x-axis or y-axis.

 

hyperbola2.jpg

hyperbola3.jpg

Figure1

 

REMARK: There is a trick that can be used to avoid memorizing the equations of the asymptotes of a hyperbola. They can be obtained, when needed, by substituting 0 for the 1 on the right side of the hyperbola equation, and then solving for y in terms of x. For example, for the hyperbola

 

we would write

or

 

take the square root

which are the equations for the asymptotes.

 

A Technique For Graphing Hyperbolas

Hyperbolas can be graphed from their standard equations using four basic steps:

roughsketch.png Rough Sketch

 

Watch example video

 

 Hyperlink to Labeling Activity 

Objective 3: Translated Hyperbolas

Standard Equations of a Hyperbola with center (h,k)

Horizontal Transverse Axis (Parallel to the x-axis)

 

hyperbola 4.jpg

Center: (h,k)

Vertices: (h-a,k) and (h+a,k)

Foci: (h-c,k) and (h+c,k)

Asymptotes:

Vertical Transverse Axis (Parallel to the y-axis)

 

hyperbola5.jpg

Center: (h,k)

Vertices: (h,k+a) and (h,k-a)

Foci: (h,k-c) and (h,k+c)

Asymptotes:

 

 

Watch example video

 

A hyperbola such as in the last example is called equilateral if a = b.

 

Watch example video

Objective 4:   Solving Applied Problems Involving Ellipse

 

Hyperbolas have many applications.   Some comets are only seen once in our solar system as they travel through the solar system on the path of a hyperbola with the sun at a focus.   On October 14, 1947, Chuck Yeager became the first person to break the sound barrier.   As an airplane moves faster than the speed of sound a cone shaped shock wave is produced. The cone intersects the ground in the shape of a hyperbola.    

 

When two rocks are simultaneously tossed into a calm pool of water, ripples move outward in the form of concentric circles.   These circles intersect in points that form a hyperbola.  

  

Another application of hyperbolas is in the long-range navigation system LORAN. This system uses transmitting stations in three locations to send out simultaneous signals to a ship or aircraft. The difference in the arrival times of the signals from one pair of transmitters is recorded on the ship or aircraft. This difference is also recorded for signals from another pair of transmitters. For each pair, a computation is performed to determine the difference in the distances from each member of the pair to the ship or aircraft. If each pair of differences is kept constant, two hyperbolas can be drawn. Each has one of the pairs of transmitters as foci, and the ship or aircraft lies on the intersection of two of their branches. See diagram below.

application.png

 

 

Activity

Quiz your concepts and vocabulary

  

Other Mathematics Resources: www.learner.org

Algebra: In Simplest Terms, look for video:

Ellipse and Hyperbola 

 

 Objective 5: Classifying Equations of Conic Sections

Equation

Type of Conic Section

Graph

Only one variable is squared, so this cannot be a cirlce, an ellipse, or a hyperbola. Find an equivalent equation by minus 4y, add 4 to both sides of the equation and then factor:

This is an equation of a parabola

parabola.png

Both variables are squared, so this cannot be a parabola. The squared terms are added, so this cannot be a hyperbola. Divide by 3 on both sides to find an equivalent equation:

This is an equation of a circle.

circle.png

Both variables are squared, so this cannot be a parabola. Add 4x2 on both sides to find an equivalent equation:

 

The squared terms are added, so this cannot be a hyperbola. The coeffiecient of x2 and y2 are not the same, so this is not a circle.

Divided by 16 on both sides to find an equivalent equation:

This is an equation of an ellipse.

ellipse.png

Both variables are squared, so this cannot be a parabola. Subtract 4y2 on both sides to find an equivalent equation:

The square terms are not added, so this cannot be a circle or an ellipse. Divided by 36 on both sides to find an equivalent equation:

This is an equation of a hyperbola.

 

hyperbola5.png

 

Review the conis section.