# Analytic Geometry

## The Hyperbola


### Learning Objectives

1. Use the Distance Formula. (IA 11.1.1)
2. Graph a hyperbola with center at (0,0). (IA 11.4.1)


### Objective 1: Use the Distance Formula. (IA 11.1.1)


### Practice Makes Perfect

Use the Distance Formula.

### Objective 2: Graph a hyperbola with center at (0,0). (IA 11.4.1)

A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Notice that, unlike the equation of an ellipse, the denominator of  is not always  and the denominator of  is not always  . 
Notice that when the  term is positive, the transverse axis is on the x-axis. When the  term is positive, the transverse axis is on the y-axis. 

### Practice Makes Perfect


 Graph a hyperbola with center at (0,0).

What do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of conic. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom. See .
Most people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.

### Locating the Vertices and Foci  of a Hyperbola

In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. See .
Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points 
  in a plane such that the difference of the distances between 
  and the foci is a positive constant.
Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances.
As with the ellipse, every hyperbola has two axes of symmetry. The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. See .
In this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the x- and y-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.

### Deriving the Equation of a Hyperbola Centered at the Origin

Let 
  and 
  be the foci of a hyperbola centered at the origin. The hyperbola is the set of all points 
  such that the difference of the distances from 
  to the foci is constant. See .
If 
  is a vertex of the hyperbola, the distance from 
  to 
  is 
  The distance from 
  to 
  is 
  The difference of the distances from the foci to the vertex is
If 
  is a point on the hyperbola, we can define the following variables:
By definition of a hyperbola, 
  is constant for any point 
  on the hyperbola. We know that the difference of these distances is 
  for the vertex 
  It follows that 
  for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula. The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.
This equation defines a hyperbola centered at the origin with vertices 
  and co-vertices 
 


### Writing Equations of Hyperbolas in Standard Form

Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.

### Hyperbolas Centered at the Origin

Reviewing the standard forms given for hyperbolas centered at 
  we see that the vertices, co-vertices, and foci are related by the equation 
  Note that this equation can also be rewritten as 
  This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.

### Hyperbolas Not Centered at the Origin 

Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated 
  units horizontally and 
  units vertically, the center of the hyperbola will be 
  This translation results in the standard form of the equation we saw previously, with 
  replaced by 
  and 
  replaced by 
 

Like hyperbolas centered at the origin, hyperbolas centered at a point 
  have vertices, co-vertices, and foci that are related by the equation 
  We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.

###  Graphing Hyperbolas Centered at the Origin

When we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify the key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the transverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form 
  for horizontal hyperbolas and the standard form 
  for vertical hyperbolas.

###  Graphing Hyperbolas Not Centered at the Origin

Graphing hyperbolas centered at a point 
  other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms 
  for horizontal hyperbolas, and 
  for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.

### Solving Applied Problems Involving Hyperbolas

As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. See . For example, a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!
The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In  we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.

### Key Equations


### Key Concepts

1. A hyperbola is the set of all points 
  in a plane such that the difference of the distances between 
  and the foci is a positive constant.
2. The standard form of a hyperbola can be used to locate its vertices and foci. See .
3. When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See  and .
4. When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See  and .
5. Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.
For the following exercises, find the equations of the asymptotes for each hyperbola.

### Graphical

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.
For the following exercises, given information about the graph of the hyperbola, find its equation.
For the following exercises, given the graph of the hyperbola, find its equation.

###  Extensions

For the following exercises, express the equation for the hyperbola as two functions, with 
  as a function of 
  Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.

###  Real-World Applications

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.