In this section, you will:
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 [link], 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.
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 [link].
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 points
in 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 [link].
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 [link].
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.
To derive the equation of an ellipse centered at the origin, we begin with the foci
and
The ellipse is the set of all points
such that the sum of the distances from
to the foci is constant, as shown in [link].
If
is a vertex of the ellipse, the distance from
to
is
The distance from
to
is
. The sum of the distances from the foci to the vertex is
If
is a point on the ellipse, then we can define the following variables:
By the definition of an ellipse,
is constant for any point
on the ellipse. We know that the sum of these distances is
for the vertex
It follows that
for 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 is
This equation defines an ellipse centered at the origin. If
the ellipse is stretched further in the horizontal direction, and if
the ellipse is stretched further in the vertical direction.
Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’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.
The standard form of the equation of an ellipse with center
and major axis on the x-axis is
where
, where
See [link]a
The standard form of the equation of an ellipse with center
and major axis on the y-axis is
where
, where
See [link]b
Note that the vertices, co-vertices, and foci are related by the equation
When 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.
Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.
and
respectively, then the major axis is the x-axis. Use the standard form
and
respectively, then the major axis is the y-axis. Use the standard form
along with the given coordinates of the vertices and foci, to solve for
and
into the standard form of the equation determined in Step 1.
What is the standard form equation of the ellipse that has vertices
and 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 are
so
and
The foci are
so
and
We know that the vertices and foci are related by the equation
Solving for
we have:
Now we need only substitute
and
into the standard form of the equation. The equation of the ellipse is
What is the standard form equation of the ellipse that has vertices
and 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 form
</math>or
</math>Similarly, the coordinates of the foci will always have the form
</math>or
</math>Knowing this, we can use
</math>and
</math>from the given points, along with the equation
</math>to find
</math></em>
Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated
units horizontally and
units vertically, the center of the ellipse will be
This translation results in the standard form of the equation we saw previously, with
replaced by
and y replaced by
The standard form of the equation of an ellipse with center
and major axis parallel to the x-axis is
where
where
See [link]a
The standard form of the equation of an ellipse with center
and major axis parallel to the y-axis is
where
where
See [link]b
Just as with ellipses centered at the origin, ellipses that are 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 equation of the ellipse in standard form when the vertices and foci are given.
Given the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.
using the midpoint formula and the given coordinates for the vertices.
by solving for the length of the major axis,
which is the distance between the given vertices.
using
and
found in Step 2, along with the given coordinates for the foci.
using the equation
and
into the standard form of the equation determined in Step 1.
What is the standard form equation of the ellipse that has vertices
and
and foci
and
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,
and
Applying the midpoint formula, we have:
Next, we find
The length of the major axis,
is bounded by the vertices. We solve for
by finding the distance between the y-coordinates of the vertices.
So
Now we find
The foci are given by
So,
and
We substitute
using either of these points to solve for
So
Next, we solve for
using the equation
Finally, we substitute the values found for
and
into the standard form equation for an ellipse:
What is the standard form equation of the ellipse that has vertices
and
and foci
and
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 form
for horizontal ellipses and
for vertical ellipses.
**Given the standard form of an equation for an ellipse centered at
sketch the graph.**
where
then
where
then
using the equation
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 on the y-axis. Therefore, the equation is in the form
where
and
It follows that:
where
Solving 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 [link].
Graph the ellipse given by the equation
Identify and label the center, vertices, co-vertices, and foci.
center:
vertices:
co-vertices:
foci:
Graph the ellipse given by the equation
Rewrite 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. Because
the major axis is on the x-axis. Therefore, the equation is in the form
where
and
It follows that:
where
Solving 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.
Graph the ellipse given by the equation
Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.
Standard form:
center:
vertices:
co-vertices:
foci:
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 forms
for horizontal ellipses and
for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.
**Given the standard form of an equation for an ellipse centered at
sketch the graph.**
where
then
where
then
using the equation
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 form
where
and
It follows that:
or
and
or
and
where
Solving for
we have:
Therefore, the coordinates of the foci are
and
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.
Graph the ellipse given by the equation
Identify and label the center, vertices, co-vertices, and foci.
Center:
vertices:
and
co-vertices:
and
foci:
and
Given the general form of an equation for an ellipse centered at (h, k), express the equation in standard form.
is in general form.
and
terms in preparation for completing the square.
where
and
are constants.
Graph the ellipse given by the equation
Identify 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 form
where
and
It follows that:
or
and
or
and
where
Solving for
we have:
Therefore, the coordinates of the foci are
and
Next we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in [link].
Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.
center:
vertices:
and
co-vertices:
and
foci:
and
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 [link]. In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.
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 [link].
so we need to find an equation of the form
where
We 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.
we have
so
and
we have
so
and
Therefore, the equation of the ellipse is
where
Solving for
we have:
The points
represent the foci. Thus, the distance between the senators is
feet.
Suppose a whispering chamber is 480 feet long and 320 feet wide.
Access these online resources for additional instruction and practice with ellipses.
Horizontal ellipse, center at origin |
Vertical ellipse, center at origin |
Horizontal ellipse, center |
Vertical ellipse, center |
in 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).
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.
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 axis
and
Endpoints of minor axis
and
Foci at
Endpoints of major axis
and
Endpoints of minor axis
Foci at
Endpoints of major axis
Endpoints of minor axis
Foci at
Endpoints of major axis
Endpoints of minor axis
Foci at
Endpoints of major axis
Endpoints of minor axis
Foci at
Endpoints of major axis
Endpoints of minor axis
Foci at
Endpoints of major axis
Endpoints of minor axis
Foci at
Endpoints of major axis
Endpoints of minor axis
Foci at
For the following exercises, find the foci for the given ellipses.
Foci
Focus
Foci
For the following exercises, graph the given ellipses, noting center, vertices, and foci.
Center
Vertices
Foci
Center
Vertices
Foci
Center
Vertices
Focus
Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.
Center
Vertices
Foci
Center
Vertices
Foci
Center
Vertices
Foci
Center
Vertices
Focus
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.
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula
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?
in a plane such that the sum of their distances from two fixed points is a constant
on the ellipse is a constant
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