Analytic Geometry

# The Hyperbola

### Learning Objectives

In this section, you will:

• Locate a hyperbola’s vertices and foci.
• Write equations of hyperbolas in standard form.
• Graph hyperbolas centered at the origin.
• Graph hyperbolas not centered at the origin.
• Solve applied problems involving hyperbolas.

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 (Figure).

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 (Figure).

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 pointsin a plane such that the difference of the distances betweenand 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 (Figure).

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

Letandbe the foci of a hyperbola centered at the origin. The hyperbola is the set of all pointssuch that the difference of the distances fromto the foci is constant. See (Figure).

Ifis a vertex of the hyperbola, the distance fromtoisThe distance fromtoisThe sum of the distances from the foci to the vertex is

Ifis a point on the hyperbola, we can define the following variables:

By definition of a hyperbola,is constant for any pointon the hyperbola. We know that the difference of these distances isfor the vertexIt follows thatfor 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 verticesand co-vertices

### Standard Forms of the Equation of a Hyperbola with Center (0,0)

The standard form of the equation of a hyperbola with centerand transverse axis on the x-axis is

where

• the length of the transverse axis is
• the coordinates of the vertices are
• the length of the conjugate axis is
• the coordinates of the co-vertices are
• the distance between the foci is where
• the coordinates of the foci are
• the equations of the asymptotes are

See (Figure)a.

The standard form of the equation of a hyperbola with centerand transverse axis on the y-axis is

where

• the length of the transverse axis is
• the coordinates of the vertices are
• the length of the conjugate axis is
• the coordinates of the co-vertices are
• the distance between the foci is where
• the coordinates of the foci are
• the equations of the asymptotes are

See (Figure)b.

Note that the vertices, co-vertices, and foci are related by the equationWhen we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci.

### How To

Given the equation of a hyperbola in standard form, locate its vertices and foci.

1. Determine whether the transverse axis lies on the x– or y-axis. Notice thatis always under the variable with the positive coefficient. So, if you set the other variable equal to zero, you can easily find the intercepts. In the case where the hyperbola is centered at the origin, the intercepts coincide with the vertices.
1. If the equation has the form then the transverse axis lies on the x-axis. The vertices are located at and the foci are located at
2. If the equation has the form then the transverse axis lies on the y-axis. The vertices are located at and the foci are located at
2. Solve forusing the equation
3. Solve forusing the equation

### Locating a Hyperbola’s Vertices and Foci

Identify the vertices and foci of the hyperbola with equation

The equation has the form so the transverse axis lies on the y-axis. The hyperbola is centered at the origin, so the vertices serve as the y-intercepts of the graph. To find the vertices, set and solve for

The foci are located atSolving for

Therefore, the vertices are located at and the foci are located at[/hidden-answer]

### Try It

Identify the vertices and foci of the hyperbola with equation

### 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 atwe see that the vertices, co-vertices, and foci are related by the equationNote that this equation can also be rewritten asThis relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.

### How To

Given the vertices and foci of a hyperbola centered at write its equation in standard form.

1. Determine whether the transverse axis lies on the x– or y-axis.
1. If the given coordinates of the vertices and foci have the formandrespectively, then the transverse axis is the x-axis. Use the standard form
2. If the given coordinates of the vertices and foci have the formandrespectively, then the transverse axis is the y-axis. Use the standard form
2. Findusing the equation
3. Substitute the values forandinto the standard form of the equation determined in Step 1.

### Finding the Equation of a Hyperbola Centered at (0,0) Given its Foci and Vertices

What is the standard form equation of the hyperbola that has verticesand foci

The vertices and foci are on the x-axis. Thus, the equation for the hyperbola will have the form

The vertices aresoand

The foci aresoand

Solving for we have

Finally, we substituteandinto the standard form of the equation,The equation of the hyperbola is as shown in (Figure).

### Try It

What is the standard form equation of the hyperbola that has verticesand foci

#### Hyperbolas Not Centered at the Origin

Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translatedunits horizontally andunits vertically, the center of the hyperbola will beThis translation results in the standard form of the equation we saw previously, withreplaced byandreplaced by

### Standard Forms of the Equation of a Hyperbola with Center (h, k)

The standard form of the equation of a hyperbola with centerand transverse axis parallel to the x-axis is

where

• the length of the transverse axis is
• the coordinates of the vertices are
• the length of the conjugate axis is
• the coordinates of the co-vertices are
• the distance between the foci iswhere
• the coordinates of the foci are

The asymptotes of the hyperbola coincide with the diagonals of the central rectangle. The length of the rectangle isand its width isThe slopes of the diagonals areand each diagonal passes through the centerUsing the point-slope formula, it is simple to show that the equations of the asymptotes areSee (Figure)a

The standard form of the equation of a hyperbola with centerand transverse axis parallel to the y-axis is

where

• the length of the transverse axis is
• the coordinates of the vertices are
• the length of the conjugate axis is
• the coordinates of the co-vertices are
• the distance between the foci iswhere
• the coordinates of the foci are

Using the reasoning above, the equations of the asymptotes areSee (Figure)b.

Like hyperbolas centered at the origin, hyperbolas centered at a pointhave vertices, co-vertices, and foci that are related by the equationWe 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.

### How To

Given the vertices and foci of a hyperbola centered atwrite its equation in standard form.

1. Determine whether the transverse axis is parallel to the x– or y-axis.
1. If the y-coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the x-axis. Use the standard form
2. If the x-coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the y-axis. Use the standard form
2. Identify the center of the hyperbola,using the midpoint formula and the given coordinates for the vertices.
3. Findby solving for the length of the transverse axis,, which is the distance between the given vertices.
4. Findusingandfound in Step 2 along with the given coordinates for the foci.
5. Solve forusing the equation
6. Substitute the values for andinto the standard form of the equation determined in Step 1.

### Finding the Equation of a Hyperbola Centered at (h, k) Given its Foci and Vertices

What is the standard form equation of the hyperbola that has vertices atandand foci atand

The y-coordinates of the vertices and foci are the same, so the transverse axis is parallel to the x-axis. Thus, the equation of the hyperbola will have the form

First, we identify the center,The center is halfway between the verticesandApplying the midpoint formula, we have

Next, we findThe length of the transverse axis,is bounded by the vertices. So, we can findby finding the distance between the x-coordinates of the vertices.

Now we need to findThe coordinates of the foci areSoandWe can use the x-coordinate from either of these points to solve forUsing the pointand substituting

Next, solve forusing the equation

Finally, substitute the values found forandinto the standard form of the equation.

### Try It

What is the standard form equation of the hyperbola that has verticesandand fociand

### 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 formfor horizontal hyperbolas and the standard formfor vertical hyperbolas.

### How To

Given a standard form equation for a hyperbola centered atsketch the graph.

1. Determine which of the standard forms applies to the given equation.
2. Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the vertices, co-vertices, and foci; and the equations for the asymptotes.
1. If the equation is in the formthen
• the transverse axis is on the x-axis
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
• the equations of the asymptotes are
2. If the equation is in the form then
• the transverse axis is on the y-axis
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
• the equations of the asymptotes are
3. Solve for the coordinates of the foci using the equation
4. Plot the vertices, co-vertices, foci, and asymptotes in the coordinate plane, and draw a smooth curve to form the hyperbola.

### Graphing a Hyperbola Centered at (0, 0) Given an Equation in Standard Form

Graph the hyperbola given by the equationIdentify and label the vertices, co-vertices, foci, and asymptotes.

The standard form that applies to the given equation isThus, the transverse axis is on the y-axis

The coordinates of the vertices are

The coordinates of the co-vertices are

The coordinates of the foci arewhereSolving forwe have

Therefore, the coordinates of the foci are

The equations of the asymptotes are

Plot and label the vertices and co-vertices, and then sketch the central rectangle. Sides of the rectangle are parallel to the axes and pass through the vertices and co-vertices. Sketch and extend the diagonals of the central rectangle to show the asymptotes. The central rectangle and asymptotes provide the framework needed to sketch an accurate graph of the hyperbola. Label the foci and asymptotes, and draw a smooth curve to form the hyperbola, as shown in (Figure).

### Try It

Graph the hyperbola given by the equationIdentify and label the vertices, co-vertices, foci, and asymptotes.

### Graphing Hyperbolas Not Centered at the Origin

Graphing hyperbolas centered at a pointother than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard formsfor horizontal hyperbolas, andfor 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.

### How To

Given a general form for a hyperbola centered at sketch the graph.

1. Convert the general form to that standard form. Determine which of the standard forms applies to the given equation.
2. Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the center, vertices, co-vertices, foci; and equations for the asymptotes.
1. If the equation is in the formthen
• the transverse axis is parallel to the x-axis
• the center is
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
• the equations of the asymptotes are
2. If the equation is in the formthen
• the transverse axis is parallel to the y-axis
• the center is
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
• the equations of the asymptotes are
3. Solve for the coordinates of the foci using the equation
4. Plot the center, vertices, co-vertices, foci, and asymptotes in the coordinate plane and draw a smooth curve to form the hyperbola.

### Graphing a Hyperbola Centered at (h, k) Given an Equation in General Form

Graph the hyperbola given by the equationIdentify and label the center, vertices, co-vertices, foci, and asymptotes.

Start by expressing the equation in standard form. Group terms that contain the same variable, and move the constant to the opposite side of the equation.

Factor the leading coefficient of each expression.

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

Rewrite as perfect squares.

Divide both sides by the constant term to place the equation in standard form.

The standard form that applies to the given equation is whereandorandThus, the transverse axis is parallel to the x-axis. It follows that:

• the center of the ellipse is
• the coordinates of the vertices areorand
• the coordinates of the co-vertices areorand
• the coordinates of the foci arewhereSolving forwe have

Therefore, the coordinates of the foci areand

The equations of the asymptotes are

Next, we plot and label the center, vertices, co-vertices, foci, and asymptotes and draw smooth curves to form the hyperbola, as shown in (Figure).

### Try It

Graph the hyperbola given by the standard form of an equationIdentify and label the center, vertices, co-vertices, foci, and asymptotes.

### 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 (Figure). 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 (Figure) we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.

### Solving Applied Problems Involving Hyperbolas

The design layout of a cooling tower is shown in (Figure). The tower stands 179.6 meters tall. The diameter of the top is 72 meters. At their closest, the sides of the tower are 60 meters apart.

Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

We are assuming the center of the tower is at the origin, so we can use the standard form of a horizontal hyperbola centered at the origin:where the branches of the hyperbola form the sides of the cooling tower. We must find the values ofandto complete the model.

First, we findRecall that the length of the transverse axis of a hyperbola isThis length is represented by the distance where the sides are closest, which is given asmeters. So,Therefore,and

To solve forwe need to substitute forandin our equation using a known point. To do this, we can use the dimensions of the tower to find some pointthat lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the y-axis bisects the tower, our x-value can be represented by the radius of the top, or 36 meters. The y-value is represented by the distance from the origin to the top, which is given as 79.6 meters. Therefore,

The sides of the tower can be modeled by the hyperbolic equation

### Try It

A design for a cooling tower project is shown in (Figure). Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

The sides of the tower can be modeled by the hyperbolic equation.[/hidden-answer]

Access these online resources for additional instruction and practice with hyperbolas.

### Key Equations

 Hyperbola, center at origin, transverse axis on x-axis Hyperbola, center at origin, transverse axis on y-axis Hyperbola, center attransverse axis parallel to x-axis Hyperbola, center attransverse axis parallel to y-axis

### Key Concepts

• A hyperbola is the set of all pointsin a plane such that the difference of the distances betweenand the foci is a positive constant.
• The standard form of a hyperbola can be used to locate its vertices and foci. See (Figure).
• When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See (Figure) and (Figure).
• 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 (Figure) and (Figure).
• 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 (Figure).

### Section Exercises

#### Verbal

Define a hyperbola in terms of its foci.

A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

What can we conclude about a hyperbola if its asymptotes intersect at the origin?

What must be true of the foci of a hyperbola?

The foci must lie on the transverse axis and be in the interior of the hyperbola.

If the transverse axis of a hyperbola is vertical, what do we know about the graph?

Where must the center of hyperbola be relative to its foci?

The center must be the midpoint of the line segment joining the foci.

#### 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.

vertices:foci:asymptotes:

vertices:foci:asymptotes:

vertices:foci:asymptotes:

vertices:foci:asymptotes:

vertices:foci:asymptotes:

vertices:foci:asymptotes:

vertices:foci:asymptotes:

vertices:foci: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.

Vertices atandand one focus at

Vertices atandand one focus at

Vertices atandand one focus at

Center:vertex:one focus:

Center:vertex:one focus:

Center:vertex:one focus:

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, withas a function ofExpress 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.

The hedge will follow the asymptotes and its closest distance to the center fountain is 5 yards.

The hedge will follow the asymptotes and its closest distance to the center fountain is 6 yards.

The hedge will follow the asymptotesand and its closest distance to the center fountain is 10 yards.

The hedge will follow the asymptotesand and its closest distance to the center fountain is 12 yards.

The hedge will follow the asymptotes and its closest distance to the center fountain is 20 yards.

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.

The object enters along a path approximated by the lineand passes within 1 au (astronomical unit) of the sun at its closest approach, so that the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line

The object enters along a path approximated by the lineand passes within 0.5 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line

The object enters along a path approximated by the lineand passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line

The object enters along a path approximated by the lineand passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line

The object enters along a path approximated by the lineand passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line

### Glossary

center of a hyperbola
the midpoint of both the transverse and conjugate axes of a hyperbola
conjugate axis
the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints
hyperbola
the set of all pointsin a plane such that the difference of the distances betweenand the foci is a positive constant
transverse axis
the axis of a hyperbola that includes the foci and has the vertices as its endpoints