Analytic Geometry

# The Ellipse

### Learning Objectives

In this section, you will:

• Write equations of ellipses in standard form.
• Graph ellipses centered at the origin.
• Graph ellipses not centered at the origin.
• Solve applied problems involving ellipses.

Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in (Figure), is such a room.[1] It is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.

### Writing Equations of Ellipses in Standard Form

A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in (Figure).

Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all pointsin a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).

We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See (Figure).

Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See (Figure).

In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x– and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.

To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.

#### Deriving the Equation of an Ellipse Centered at the Origin

To derive the equation of an ellipse centered at the origin, we begin with the fociandThe ellipse is the set of all pointssuch that the sum of the distances fromto the foci is constant, as shown in (Figure).

Ifis a vertex of the ellipse, the distance fromtoisThe distance fromtois. The sum of the distances from the foci to the vertex is

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

By the definition of an ellipse,is constant for any pointon the ellipse. We know that the sum of these distances isfor the vertexIt follows thatfor any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.

Thus, the standard equation of an ellipse isThis equation defines an ellipse centered at the origin. Ifthe ellipse is stretched further in the horizontal direction, and if the ellipse is stretched further in the vertical direction.

#### Writing Equations of Ellipses Centered at the Origin in Standard Form

Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations weâ€™ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.

The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.

### Standard Forms of the Equation of an Ellipse with Center (0,0)

The standard form of the equation of an ellipse with centerand major axis on the x-axis is

where

• the length of the major axis is
• the coordinates of the vertices are
• the length of the minor axis is
• the coordinates of the co-vertices are
• the coordinates of the foci are, whereSee (Figure)a

The standard form of the equation of an ellipse with centerand major axis on the y-axis is

where

• the length of the major axis is
• the coordinates of the vertices are
• the length of the minor axis is
• the coordinates of the co-vertices are
• the coordinates of the foci are, whereSee (Figure)b

Note that the vertices, co-vertices, and foci are related by the equationWhen we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.

### How To

Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.

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

### Writing the Equation of an Ellipse Centered at the Origin in Standard Form

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

The foci are on the x-axis, so the major axis is the x-axis. Thus, the equation will have the form

The vertices aresoand

The foci aresoand

We know that the vertices and foci are related by the equationSolving for we have:

Now we need only substituteandinto the standard form of the equation. The equation of the ellipse is[/hidden-answer]

### Try It

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

Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?

Yes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the formorSimilarly, the coordinates of the foci will always have the formorKnowing this, we can useandfrom the given points, along with the equationto find

#### Writing Equations of Ellipses Not Centered at the Origin

Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translatedunits horizontally andunits vertically, the center of the ellipse will beThis translation results in the standard form of the equation we saw previously, withreplaced byand y replaced by

### Standard Forms of the Equation of an Ellipse with Center (h, k)

The standard form of the equation of an ellipse with centerand major axis parallel to the x-axis is

where

• the length of the major axis is
• the coordinates of the vertices are
• the length of the minor axis is
• the coordinates of the co-vertices are
• the coordinates of the foci arewhereSee (Figure)a

The standard form of the equation of an ellipse with centerand major axis parallel to the y-axis is

where

• the length of the major axis is
• the coordinates of the vertices are
• the length of the minor axis is
• the coordinates of the co-vertices are
• the coordinates of the foci arewhereSee (Figure)b

Just as with ellipses centered at the origin, ellipses that are 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 equation of the ellipse in standard form when the vertices and foci are given.

### How To

Given the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.

1. Determine whether the major 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 major 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 major axis is parallel to the y-axis. Use the standard form
2. Identify the center of the ellipseusing the midpoint formula and the given coordinates for the vertices.
3. Findby solving for the length of the major axis, which is the distance between the given vertices.
4. Findusingand found 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.

### Writing the Equation of an Ellipse Centered at a Point Other Than the Origin

What is the standard form equation of the ellipse that has verticesand

and fociand

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

First, we identify the center,The center is halfway between the vertices,andApplying the midpoint formula, we have:

Next, we findThe length of the major axis, is bounded by the vertices. We solve forby finding the distance between the y-coordinates of the vertices.

So

Now we findThe foci are given bySo,andWe substituteusing either of these points to solve for

So

Next, we solve forusing the equation

Finally, we substitute the values found for andinto the standard form equation for an ellipse:

### Try It

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

### Graphing Ellipses Centered at the Origin

Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard formfor horizontal ellipses andfor vertical ellipses.

### How To

Given the standard form of an equation for an ellipse centered at sketch the graph.

1. Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.
1. If the equation is in the formwherethen
• the major axis is the x-axis
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
2. If the equation is in the formwherethen
• the major axis is the y-axis
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
2. Solve forusing the equation
3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.

### Graphing an Ellipse Centered at the Origin

Graph the ellipse given by the equation,Identify and label the center, vertices, co-vertices, and foci.

First, we determine the position of the major axis. Becausethe major axis is on the y-axis. Therefore, the equation is in the formwhereandIt follows that:

• the center of the ellipse is
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci arewhereSolving for we have:

Therefore, the coordinates of the foci are

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See (Figure).

### Try It

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

center:vertices:co-vertices:foci:

### Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form

Graph the ellipse given by the equationRewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

First, use algebra to rewrite the equation in standard form.

Next, we determine the position of the major axis. Becausethe major axis is on the x-axis. Therefore, the equation is in the formwhereandIt follows that:

• the center of the ellipse is
• 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

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

### Try It

Graph the ellipse given by the equationRewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

Standard form:center:vertices:co-vertices:foci:

### Graphing Ellipses Not Centered at the Origin

When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,we use the standard formsfor horizontal ellipses andfor vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.

### How To

Given the standard form of an equation for an ellipse centered at sketch the graph.

1. Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.
1. If the equation is in the formwherethen
• the center is
• the major axis is parallel to the x-axis
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
2. If the equation is in the formwherethen
• the center is
• the major axis is parallel to the y-axis
• the coordinates of the vertices are
• the coordinates of the co-vertices are
• the coordinates of the foci are
2. Solve forusing the equation
3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.

### Graphing an Ellipse Centered at (h, k)

Graph the ellipse given by the equation,Identify and label the center, vertices, co-vertices, and foci.

First, we determine the position of the major axis. Because the major axis is parallel to the y-axis. Therefore, the equation is in the formwhereandIt follows that:

• the center of the ellipse is
• the coordinates of the vertices are orand
• the coordinates of the co-vertices are orand
• the coordinates of the foci arewhereSolving forwe have:

Therefore, the coordinates of the foci areand

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

### Try It

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

Center:vertices:andco-vertices:andfoci:and

### How To

Given the general form of an equation for an ellipse centered at (h, k), express the equation in standard form.

1. Recognize that an ellipse described by an equation in the formis in general form.
2. Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.
3. Factor out the coefficients of theandterms in preparation for completing the square.
4. Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant, where andare constants.
5. Divide both sides of the equation by the constant term to express the equation in standard form.

### Graphing an Ellipse Centered at (h, k) by First Writing It in Standard Form

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

We must begin by rewriting the equation in standard form.

Group terms that contain the same variable, and move the constant to the opposite side of the equation.

Factor out the coefficients of the squared terms.

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.

Now that the equation is in standard form, we can determine the position of the major axis. Because the major axis is parallel to the x-axis. Therefore, the equation is in the formwhereandIt 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

Next we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in (Figure).

### Try It

Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.

center:vertices:andco-vertices:andfoci:and

### Solving Applied Problems Involving Ellipses

Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See (Figure). In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the fociâ€”about 43 feet apartâ€”can hear each other whisper.

### Locating the Foci of a Whispering Chamber

The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in (Figure).

1. What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point
2. If two senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.

1. We are assuming a horizontal ellipse with center so we need to find an equation of the formwhereWe know that the length of the major axis,is longer than the length of the minor axis,So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis.
• Solving for we have so and
• Solving for we have so and

Therefore, the equation of the ellipse is

2. To find the distance between the senators, we must find the distance between the foci,whereSolving forwe have:

The pointsrepresent the foci. Thus, the distance between the senators isfeet.[/hidden-answer]

### Try It

Suppose a whispering chamber is 480 feet long and 320 feet wide.

1. What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point
2. If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.

1. The people are standing 358 feet apart.

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

### Key Equations

 Horizontal ellipse, center at origin Vertical ellipse, center at origin Horizontal ellipse, center Vertical ellipse, center

### Key Concepts

• An ellipse is the set of all pointsin a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
• When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See (Figure) and (Figure).
• When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See (Figure) and (Figure).
• When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See (Figure) and (Figure).
• Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See (Figure).

### Section Exercises

#### Verbal

Define an ellipse in terms of its foci.

An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

Where must the foci of an ellipse lie?

What special case of the ellipse do we have when the major and minor axis are of the same length?

This special case would be a circle.

For the special case mentioned above, what would be true about the foci of that ellipse?

What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y-axis?

It is symmetric about the x-axis, y-axis, and the origin.

#### Algebraic

For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.

yes;

yes;

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

Endpoints of major axisandEndpoints of minor axisandFoci at

Endpoints of major axisandEndpoints of minor axisFoci at

Endpoints of major axisEndpoints of minor axisFoci at

Endpoints of major axisEndpoints of minor axisFoci at

Endpoints of major axisEndpoints of minor axisFoci at

Endpoints of major axisEndpoints of minor axisFoci at

Endpoints of major axisEndpoints of minor axisFoci at

Endpoints of major axisEndpoints of minor axisFoci at

For the following exercises, find the foci for the given ellipses.

Foci

Foci

#### Graphical

For the following exercises, graph the given ellipses, noting center, vertices, and foci.

CenterVerticesFoci

CenterVerticesFoci

CenterVerticesFocus

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

CenterVerticesFoci

CenterVerticesFoci

CenterVerticesFoci

CenterVerticesFocus

For the following exercises, use the given information about the graph of each ellipse to determine its equation.

Center at the origin, symmetric with respect to the x– and y-axes, focus at and point on graph

Center at the origin, symmetric with respect to the x– and y-axes, focus at and point on graph

Center at the origin, symmetric with respect to the x– and y-axes, focus at and major axis is twice as long as minor axis.

Center; vertex; one focus:.

Center; vertex; one focus:

Center; vertex; one focus:

For the following exercises, given the graph of the ellipse, determine its equation.

#### Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula

#### Real-World Applications

Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.

Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of the height.

An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of the arch at a distance of 4 feet from the center.

An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.

. Distance = 17.32 feet

A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.

A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.

Approximately 51.96 feet

A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and height at the center of the gallery?

### Glossary

center of an ellipse
the midpoint of both the major and minor axes
conic section
any shape resulting from the intersection of a right circular cone with a plane
ellipse
the set of all pointsin a plane such that the sum of their distances from two fixed points is a constant
foci
plural of focus
focus (of an ellipse)
one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any pointon the ellipse is a constant
major axis
the longer of the two axes of an ellipse
minor axis
the shorter of the two axes of an ellipse

1. Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014.