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B. The Equation of an Ellipse

B. The Equation of an Ellipse

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Section 8.2 The Circle and the Ellipse



1x Ϫ 32 2



WORTHY OF NOTE

If you point a flashlight at the floor

keeping it perpendicular to the

ground, a circle is formed with the

bulb pointing directly at the center

and every point along the outer

edge of the beam an equal

distance from this center. If you

hold the flashlight at an angle, the

circle is elongated and becomes an

ellipse, with the bulb pointing

directly at one focus.



42



ϩ



1Ϫ2 ϩ 22 2



32

1x Ϫ 32 2



ϭ1



ϩ0ϭ1

42

1x Ϫ 32 2 ϭ 16



x Ϫ 3 ϭ Ϯ4



xϭ3Ϯ4

x ϭ 7 and x ϭ Ϫ1



substitute Ϫ2 for y



simplify

multiply by 42 ϭ 16

property of square roots

add 3



This shows the horizontal distance from the center to the graph is still a ϭ 4, and

the points (Ϫ1, Ϫ22 and (7, Ϫ2) are on the graph (see Figure 8.12). Similarly, for

x ϭ 3 we have 1y ϩ 22 2 ϭ 9, giving y ϭ Ϫ5 and y ϭ 1, and showing the vertical distance from the center to the graph is now b ϭ 3, with points (3, 1) and (3, Ϫ5) on the

graph. Using this information to sketch the curve reveals the “circle” is elongated and

has become a horizontal ellipse.

For this ellipse, the line segment through the center, parallel the x-axis, and with

endpoints on the ellipse is called the major axis, with the endpoints of the major axis

called vertices. The segment perpendicular to and bisecting the major axis (with its

endpoints on the ellipse) is called the minor axis, as shown in Figure 8.13.

Figure 8.12

y

3



Figure 8.13



(3, 1)



Major axis

bϭ3



Ϫ2



(Ϫ1, Ϫ2)



8



aϭ4



(3, Ϫ2)

Ϫ5



a



x



(7, Ϫ2)



b

Vertex



Vertex



Ellipse

(3, Ϫ5)



The case where

a>b



Minor axis



• If a2 7 b2, the major axis is horizontal (parallel to the x-axis) with length 2a, and

the minor axis is vertical with length 2b (see Example 3).

• If a2 6 b2 the major axis is vertical (parallel to the y-axis) with length 2b, and the

minor axis is horizontal with length 2a (see Example 4).

Generalizing this observation we obtain the equation of an ellipse in standard form.

The Equation of an Ellipse in Standard Form

Given

If a



1x Ϫ h2 2



ϩ



1y Ϫ k2 2



ϭ 1.

a2

b2

b the equation represents the graph of an ellipse with center at (h, k).

• ͿaͿ gives the horizontal distance from center to graph.

• ͿbͿ gives the vertical distance from center to graph.



Finally, note the line segment from center to vertex is called the semimajor axis, with

the perpendicular line segment from center to graph called the semiminor axis.

EXAMPLE 3







Graphing a Horizontal Ellipse

Sketch the graph of the ellipse defined by

1y ϩ 12 2

1x Ϫ 22 2

ϩ

ϭ 1.

25

9



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CHAPTER 8 Analytic Geometry and the Conic Sections



Solution







Noting a b, we have an ellipse with

center 1h, k2 ϭ 12, Ϫ12. The horizontal

distance from the center to the graph is

a ϭ 5, and the vertical distance from the

center to the graph is b ϭ 3. After plotting

the corresponding points and connecting

them with a smooth curve, we obtain the

graph shown.



y

Ellipse

(2, 2)



(Ϫ3, Ϫ1)



bϭ3

aϭ5

(2, Ϫ1)



x



(7, Ϫ1)



(2, Ϫ4)



Now try Exercises 19 through 24



As with the circle, the equation of an ellipse can be given in polynomial form, and

here our knowledge of circles is helpful. For the equation 25x2 ϩ 4y2 ϭ 100, we

know the graph cannot be a circle since the coefficients are unequal, and the center of

the graph must be at the origin since h ϭ k ϭ 0. To actually draw the graph, we convert the equation to standard form. Note that a circle whose center is at (0, 0) is called

a central circle, and an ellipse with center at (0, 0) is called a central ellipse.



WORTHY OF NOTE

In general, for the equation

Ax2 ϩ By2 ϭ F (A, B, F 7 0), the

equation represents a circle if

A ϭ B, and an ellipse if A B.



EXAMPLE 4











Graphing a Vertical Ellipse

For 25x2 ϩ 4y2 ϭ 100,

a. Write the equation in standard form and identify the center and the values of a

and b.

b. Identify the major and minor axes and name the vertices.

c. Sketch the graph.

d. Graph the relation on a graphing calculator using a “friendly” window, then

use the TRACE feature to find four additional points on the graph whose

coordinates are rational.



Solution







The coefficients of x2 and y2 are unequal, and 25, 4, and 100 have like signs. The

equation represents an ellipse with center at (0, 0). To obtain standard form:

a. 25x2 ϩ 4y2 ϭ 100 given equation

4y2

25x2

ϩ

ϭ1

divide by 100

100

100

y2

x2

ϩ

ϭ1

standard form

4

25

y2

x2

ϩ

ϭ1

write denominators in squared form; a ϭ 2, b ϭ 5

22

52

b. The result shows a ϭ 2 and b ϭ 5, indicating the major axis will be vertical and

the minor axis will be horizontal. With the center at the origin, the x-intercepts will

Figure 8.14

be 1Ϫ2, 02 and (2, 0),

y

with the vertices (and

Vertical ellipse

(0, 5)

y-intercepts) at

Center at (0, 0)

10, Ϫ52 and (0, 5).

bϭ5

Endpoints of major axis (vertices)

c. Plotting these

(0, Ϫ5) and (0, 5)

intercepts and

(Ϫ2, 0)

(2, 0)

Endpoints of minor axis

x

sketching the ellipse

(Ϫ2, 0) and (2, 0)

a

ϭ

2

results in the graph

Length of major axis 2b: 2(5) ϭ 10

shown in

Length of minor axis 2a: 2(2) ϭ 4

Figure 8.14.

(0, Ϫ5)



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Section 8.2 The Circle and the Ellipse



d. As with the circle, we begin by solving for y.

25x2 ϩ 4y2 ϭ 100

4y2 ϭ 100 Ϫ 25x2

100 Ϫ 25x2

y2 ϭ

4

100 Ϫ 25x2

yϭϮ

B

4

Y1 ϭ ϩ



719



original equation

isolate term containing y

divide by 4



take square roots



100 Ϫ 25X2

100 Ϫ 25X2

, Y2 ϭ Ϫ

B

B

4

4

Figure 8.15

6.2



The graph is shown in Figure 8.15, where

we note that (1.6, 3) is a point on the graph.

Due to the symmetry of the ellipse,

1Ϫ1.6, 32 , 1Ϫ1.6, Ϫ32 , and 11.6, Ϫ32 are

also on the graph.



Ϫ9.4



9.4



Ϫ6.2



Now try Exercises 25 through 36

WORTHY OF NOTE

After writing the equation in

standard form, it is possible to end

up with a constant that is zero or

negative. In the first case, the graph

is a single point. In the second

case, no graph is possible since

roots of the equation will be

complex numbers. These are called

degenerate cases. See Exercise 84.



EXAMPLE 5







If the center of the ellipse is not

at the origin, the polynomial form

has additional linear terms and we

must first complete the square in x

and y, then write the equation in standard form to sketch the graph (see the

Reinforcing Basic Concepts feature

for more on completing the square).

Figure 8.16 illustrates how the central ellipse and the shifted ellipse are

related.







Figure 8.16

y

Ellipse with center

at (h, k)

k



(h, k)



Central

ellipse (0, b)

(Ϫa, 0)

(a, 0)

(0, 0)

(0, Ϫb)



All points shift

h units horizontally,

k units vertically,

opposite the sign

(x Ϫ h)2 (y Ϫ k)2

ϭ1

ϩ

a2

b2

a2 Ͼ b2



x

h

x2

y2

ϩ 2 ϭ1

a2

b

aϾb



Completing the Square to Graph an Ellipse

Sketch the graph of 25x2 ϩ 4y2 ϩ 150x Ϫ 16y ϩ 141 ϭ 0, then state the domain and range of

the relation.



Solution







The coefficients of x2 and y2 are unequal and have like signs, and we assume the equation

represents an ellipse but wait until we have the factored form to be certain (it could be a

degenerate ellipse).

25x2 ϩ 4y2 ϩ 150x Ϫ 16y ϩ 141 ϭ 0

25x2 ϩ 150x ϩ 4y2 Ϫ 16y ϭ Ϫ141

2

251x ϩ 6x ϩ __ 2 ϩ 41y2 Ϫ 4y ϩ __ 2 ϭ Ϫ141

251x2 ϩ 6x ϩ 92 ϩ 41y2 Ϫ 4y ϩ 42 ϭ Ϫ141 ϩ 225 ϩ 16

c



c



adds 25192 ϭ 225



c



c

adds 4142 ϭ 16



add 225 ϩ 16 to right



given equation (polynomial form)

group like terms; subtract 141

factor out leading coefficient from each group

complete the square



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CHAPTER 8 Analytic Geometry and the Conic Sections



251x ϩ 32 2 ϩ 41y Ϫ 22 2 ϭ 100

41y Ϫ 22 2

251x ϩ 32 2

100

ϩ

ϭ

100

100

100

1y Ϫ 22 2

1x ϩ 32 2

ϩ

ϭ1

4

25

1x ϩ 32 2

1y Ϫ 22 2

ϩ

ϭ1

22

52

The result is a vertical ellipse with

center at 1Ϫ3, 22, with a ϭ 2 and

b ϭ 5. The vertices are a vertical

distance of 5 units from center,

and the endpoints of the minor

axis are a horizontal distance of

2 units from center. Note this is

the same ellipse as in Example 4,

but shifted 3 units left and 2 up.

The domain of this relation is

x ʦ 3 Ϫ5, Ϫ1 4 , and the range is

y ʦ 3 Ϫ3, 7 4.



factor

divide both sides by 100



simplify (standard form)



write denominators in squared form



(Ϫ3, 7)



y



Vertical ellipse

Center at (Ϫ3, 2)



(Ϫ5, 2)



B. You’ve just seen how

we can use the equation of an

ellipse to graph central and

noncentral ellipses



(Ϫ3, 2)



Endpoints of major axis (vertices)

(Ϫ3, Ϫ3) and (Ϫ3, 7)

(Ϫ1, 2)



Endpoints of minor axis

(Ϫ5, 2) and (Ϫ1, 2)

x Length of major axis 2b: 2(5) ϭ 10

Length of minor axis 2a: 2(2) ϭ 4



(Ϫ3, Ϫ3)



Now try Exercises 37 through 44







C. The Foci of an Ellipse

In Section 8.1, we noted that an ellipse could also be defined in terms of two special

points called the foci. The Museum of Science and Industry in Chicago, Illinois

(http://www.msichicago.org), has a permanent exhibit called the Whispering Gallery.

The construction of the room is based on some of the reflective properties of an ellipse.

If two people stand at designated points in the room and one of them whispers very

softly, the other person can hear the whisper quite clearly—even though they are over

40 ft apart! The point where each person stands is a focus of an ellipse. This reflective

property also applies to light and radiation, giving the ellipse some powerful applications in science, medicine, acoustics, and other areas. To understand and appreciate

these applications, we introduce the analytic definition of an ellipse.

WORTHY OF NOTE

You can easily draw an ellipse that

satisfies the definition. Press two

pushpins (these form the foci of the

ellipse) halfway down into a piece

of heavy cardboard about 6 in.

apart. Take an 8-in. piece of string

and loop each end around the pins.

Use a pencil to draw the string taut

and keep it taut as you move the

pencil in a circular motion—and

the result is an ellipse! A different

length of string or a different

distance between the foci will

produce a different ellipse.



Definition of an Ellipse

Given two fixed points f1 and f2 in a plane, an ellipse

is the set of all points (x, y) where the distance from

f1 to (x, y) added to the distance from f2 to (x, y)

remains constant.



y

P(x, y)

d1



d1 ϩ d2 ϭ k

The fixed points f1 and f2 are called the foci of the

ellipse, and the points P(x, y) are on the graph of the

ellipse.



f1



d2

f2



x



d1 ϩ d2 ϭ k



6 in.

3 in.



5 in.



To find the equation of an ellipse in terms of a and b we combine the definition

just given with the distance formula. Consider the ellipse shown in Figure 8.17 (for



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Section 8.2 The Circle and the Ellipse



Figure 8.17

y

(0, b)

P(x, y)

(a, 0)

x



(Ϫa, 0)

(Ϫc, 0)



(c, 0)



calculating ease we use a central ellipse). Note the vertices have coordinates 1Ϫa, 02 and

(a, 0), and the endpoints of the minor axis have coordinates 10, Ϫb2 and (0, b) as

before. It is customary to assign foci the coordinates f1 S 1Ϫc, 02 and f2 S 1c, 02. We

can calculate the distance between (c, 0) and any point P(x, y) on the ellipse using the

distance formula:

21x Ϫ c2 2 ϩ 1y Ϫ 02 2



Likewise the distance between 1Ϫc, 02 and any point (x, y) is

21x ϩ c2 2 ϩ 1y Ϫ 02 2



(0, Ϫb)



According to the definition, the sum must be constant:

21x Ϫ c2 2 ϩ y2 ϩ 21x ϩ c2 2 ϩ y2 ϭ k

EXAMPLE 6







Finding the Value of k from the Definition of an Ellipse

Use the definition of an ellipse and the diagram given to determine the constant k

used for this ellipse (also see the following Worthy of Note). Note that

a ϭ 5, b ϭ 3, and c ϭ 4.

y

(0, 3)



P(3, 2.4)



(Ϫ5, 0)

(Ϫ4, 0)



(4, 0)



(5, 0)

x



(0, Ϫ3)



Solution







21x Ϫ c2 2 ϩ 1y Ϫ 02 2 ϩ 21x ϩ c2 2 ϩ 1y Ϫ 02 2 ϭ k



213 Ϫ 42 ϩ 12.4 Ϫ 02 ϩ 213 ϩ 42 ϩ 12.4 Ϫ 02 ϭ k

2



WORTHY OF NOTE

Note that if the foci are coincident

(both at the origin) the “ellipse” will

k

actually be a circle with radius ;

2

2x2 ϩ y2 ϩ 2x2 ϩ y2 ϭ k leads to

k2

x2 ϩ y2 ϭ . In Example 6 we

4

10

ϭ 5, and if

found k ϭ 10, giving

2

we used the “string” to draw the

circle, the pencil would be 5 units

from the center, creating a circle of

radius 5.



2



2



2



21Ϫ12 ϩ 2.4 ϩ 27 ϩ 2.4 ϭ k

16.76 ϩ 154.76 ϭ k

2.6 ϩ 7.4 ϭ k

10 ϭ k

The constant value for this ellipse is 10 units.

2



2



2



2



given

substitute

add

simplify radicals

compute square roots

result



Now try Exercises 45 through 48

In Example 6, the sum of the distances

could also be found by moving the point (x, y)

to the location of a vertex (a, 0), then using

the symmetry of the ellipse. The sum is identical to the length of the major axis, since the

overlapping part of the string from (c, 0) to

(a, 0) is the same length as from (Ϫa, 0) to

(Ϫc, 0) (see Figure 8.18). This shows the

constant k is equal to 2a regardless of the distance between foci.

As we noted, the result is







Figure 8.18

y

d1 ϩ d2 ϭ 2a

d1



d2



(Ϫa, 0)

(Ϫc, 0)



21x Ϫ c2 2 ϩ y2 ϩ 21x ϩ c2 2 ϩ y2 ϭ 2a



(c, 0)



(a, 0)

x



These two segments

are equal

substitute 2a for k



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B. The Equation of an Ellipse

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