Chapter 10
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Chapter 10 Resource Masters
1 2 3 4 5 6 7 8 9 10 XXX 11 10 09 08 07 06 05 04
Contents
Vocabulary Builder . . . . . . . . . . . . . . . vii-viii Lesson 10-7
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 435
Lesson 10-1 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 417 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Lesson 10-8
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 438
Lesson 10-2 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 420 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 440
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Chapter 10 Assessment
Chapter 10 Test, Form 1A . . . . . . . . . . . 441-442
Lesson 10-3 Chapter 10 Test, Form 1B . . . . . . . . . . . 443-444
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 423 Chapter 10 Test, Form 1C . . . . . . . . . . . 445-446
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Chapter 10 Test, Form 2A . . . . . . . . . . . 447-448
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Chapter 10 Test, Form 2B . . . . . . . . . . . 449-450
Chapter 10 Test, Form 2C . . . . . . . . . . . 451-452
Lesson 10-4 Chapter 10 Extended Response
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 426
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Assessment . . . . . . . . . . . . . . . . . . . . . . . 453
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Chapter 10 Mid-Chapter Test . . . . . . . . . . . . 454
Chapter 10 Quizzes A & B . . . . . . . . . . . . . . 455
Lesson 10-5 Chapter 10 Quizzes C & D. . . . . . . . . . . . . . 456
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 429 Chapter 10 SAT and ACT Practice . . . . 457-458
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Chapter 10 Cumulative Review . . . . . . . . . . 459
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 431
SAT and ACT Practice Answer Sheet,
Lesson 10-6 10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 432
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 SAT and ACT Practice Answer Sheet,
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 434 20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A17
© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts
A Teacher’s Guide to Using the
Chapter 10 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the
resources you use most often. The Chapter 10 Resource Masters include the core
materials needed for Chapter 10. These materials include worksheets, extensions,
and assessment options. The answers for these pages appear at the back of this
booklet.
All of the materials found in this booklet are included for viewing and printing in
the Advanced Mathematical Concepts TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-viii include a Practice There is one master for each lesson.
student study tool that presents the key These problems more closely follow the
vocabulary terms from the chapter. Students are structure of the Practice section of the Student
to record definitions and/or examples for each Edition exercises. These exercises are of
term. You may suggest that students highlight or average difficulty.
star the terms with which they are not familiar.
When to Use These provide additional
When to Use Give these pages to students practice options or may be used as homework
before beginning Lesson 10-1. Remind them to for second day teaching of the lesson.
add definitions and examples as they complete
each lesson.
Study Guide There is one Study Guide Enrichment There is one master for each
master for each lesson. lesson. These activities may extend the concepts
in the lesson, offer a historical or multicultural
When to Use Use these masters as look at the concepts, or widen students’
reteaching activities for students who need perspectives on the mathematics they are
additional reinforcement. These pages can also learning. These are not written exclusively
be used in conjunction with the Student Edition for honors students, but are accessible for use
as an instructional tool for those students who with all levels of students.
have been absent.
When to Use These may be used as extra
credit, short-term projects, or as activities for
days when class periods are shortened.
© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts
Assessment Options Intermediate Assessment
• A Mid-Chapter Test provides an option to
The assessment section of the Chapter 10
Resources Masters offers a wide range of assess the first half of the chapter. It is
assessment tools for intermediate and final composed of free-response questions.
assessment. The following lists describe each
assessment master and its intended use. • Four free-response quizzes are included to
offer assessment at appropriate intervals in
the chapter.
Chapter Assessments Continuing Assessment
• The SAT and ACT Practice offers
Chapter Tests
• Forms 1A, 1B, and 1C Form 1 tests contain continuing review of concepts in various
formats, which may appear on standardized
multiple-choice questions. Form 1A is tests that they may encounter. This practice
intended for use with honors-level students, includes multiple-choice, quantitative-
Form 1B is intended for use with average- comparison, and grid-in questions. Bubble-
level students, and Form 1C is intended for in and grid-in answer sections are provided
use with basic-level students. These tests on the master.
are similar in format to offer comparable
testing situations. • The Cumulative Review provides students
an opportunity to reinforce and retain skills
• Forms 2A, 2B, and 2C Form 2 tests are as they proceed through their study of
composed of free-response questions. Form advanced mathematics. It can also be used
2A is intended for use with honors-level as a test. The master includes free-response
students, Form 2B is intended for use with questions.
average-level students, and Form 2C is
intended for use with basic-level students. Answers
These tests are similar in format to offer • Page A1 is an answer sheet for the SAT and
comparable testing situations.
ACT Practice questions that appear in the
All of the above tests include a challenging Student Edition on page 693. Page A2 is an
Bonus question. answer sheet for the SAT and ACT Practice
master. These improve students’ familiarity
• The Extended Response Assessment with the answer formats they may
includes performance assessment tasks that encounter in test taking.
are suitable for all students. A scoring
rubric is included for evaluation guidelines. • The answers for the lesson-by-lesson
Sample answers are provided for masters are provided as reduced pages with
assessment. answers appearing in red.
• Full-size answer keys are provided for the
assessment options in this booklet.
© Glencoe/McGraw-Hill v Advanced Mathematical Concepts
Chapter 10 Leveled Worksheets
Glencoe’s leveled worksheets are helpful for meeting the needs of every
student in a variety of ways. These worksheets, many of which are found
in the FAST FILE Chapter Resource Masters, are shown in the chart
below.
• Study Guide masters provide worked-out examples as well as practice
problems.
• Each chapter’s Vocabulary Builder master provides students the
opportunity to write out key concepts and definitions in their own
words.
• Practice masters provide average-level problems for students who
are moving at a regular pace.
• Enrichment masters offer students the opportunity to extend their
learning.
Five Different Options to Meet the Needs of
Every Student in a Variety of Ways
primarily skills AVERAGE ADVANCED
primarily concepts
primarily applications
BASIC
1 Study Guide
2 Vocabulary Builder
3 Parent and Student Study Guide (online)
4 Practice
5 Enrichment
© Glencoe/McGraw-Hill vi Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Reading to Learn Mathematics
Vocabulary Builder
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 10.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term.
Vocabulary Term Found Definition/Description/Example
analytic geometry on Page
asymptotes
axis of symmetry
center
conic section
conjugate axis
degenerate conic
directrix
eccentricity
ellipse
© Glencoe/McGraw-Hill (continued on the next page)
vii Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term Found Definition/Description/Example
equilateral hyperbola on Page
focus
hyperbola
locus
major axis
minor axis
rectangular hyperbola
semi-major axis
semi-minor axis
transverse axis
vertex
© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts
10-1 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Introduction to Analytic Geometry
Example 1 Find the distance between points at (Ϫ2, 2) and
(5, Ϫ4). Then find the midpoint of the segment
that has endpoints at the given coordinates.
d ϭ ͙ෆ(x2ෆෆϪxෆ1ෆ)ෆ2 ϩෆෆ(ෆy2ෆෆϪෆyෆ1)ෆ2 Distance Formula
Let (x1, y1) ϭ (Ϫ2, 2) and
d ϭ ͙ෆ[5ෆϪෆ(ෆϪෆ2)ෆ]2ෆϩෆෆ[(ෆϪ4ෆ)ෆෆϪෆ2]ෆ2 (x2, y2) ϭ (5, Ϫ4).
d ϭ ͙ෆ72ෆϩෆෆ(Ϫෆෆ6)ෆ2 or ͙ෆ8ෆ5
midpoint ϭ ᎏx1 ϩ2ᎏx2 , ᎏy1 ϩ2ᎏy2 Midpoint Formula
The midpoint is at ᎏϪ22ᎏϩ 5 , ᎏ2 ϩ2(ᎏϪ4) or ᎏ23ᎏ, Ϫ1 .
Example 2 Determine whether quadrilateral ABCD with
vertices A(1, 1), B(0, Ϫ1), C(Ϫ2, 0), and D(Ϫ1, 2) is
a parallelogram.
First, graph the figure.
To determine if DA ͉͉ CB, find the slopes of DA and CB.
slope of DA slope of CB
m ϭ ᎏyx22 ϪϪᎏxy11 Slope formula m ϭ ᎏxy22 ϪϪᎏyx11 Slope formula
C(Ϫ2, 0) and B(0, Ϫ1)
ϭ ᎏ1 Ϫᎏ2 D(Ϫ1, 2) and A(1, 1) ϭ ᎏϪ1 Ϫᎏ0
1 Ϫ (Ϫ1) 0 Ϫ (Ϫ2)
ϭ Ϫᎏ21ᎏ ϭ Ϫᎏ12ᎏ
Their slopes are equal. Therefore, DA ͉͉ CB.
fTionddeDteArmanindeCiBf D. A Х CB, use the distance formula to
DA ϭ ͙ෆ[1ෆϪෆ(ෆϪෆ1ෆ)]2ෆෆϩෆ(ෆ1ෆϪෆ2)ෆ2 CB ϭ ͙[ෆ0ෆϪෆ(ෆϪෆ2ෆ)]2ෆෆϩෆ(ෆϪෆ1ෆϪෆ0ෆ)2
ϭ ͙ෆ5 ϭ ͙ෆ5
The measures of DA and CB are equal. Therefore, DA Х CB.
Since DA ͉͉ CB and DA Х CB, quadrilateral ABCD is a
parallelogram.
© Glencoe/McGraw-Hill 417 Advanced Mathematical Concepts
10-1 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Introduction to Analytic Geometry
Find the distance between each pair of points with the given
coordinates. Then find the midpoint of the segment that has
endpoints at the given coordinates.
1. (Ϫ2, 1), (3, 4) 2. (1, 1), (9, 7)
3. (3, Ϫ4),(5, 2) 4. (Ϫ1, 2), (5, 4)
5. (Ϫ7, Ϫ4), (2, 8) 6. (Ϫ4, 10), (4, Ϫ5)
Determine whether the quadrilateral having vertices with the
given coordinates is a parallelogram.
7. (4, 4), (2, Ϫ2), (Ϫ5, Ϫ1), (Ϫ3, 5) 8. (3, 5), (Ϫ1, 1), (Ϫ6, 2), (Ϫ3, 7)
9. (4, Ϫ1), (2, Ϫ5), (Ϫ3, Ϫ3), (Ϫ1, 1) 10. (2, 6), (1, 2), (Ϫ4, 4), (Ϫ3, 9)
11. Hiking Jenna and Maria are hiking to a campsite located at
(2, 1) on a map grid, where each side of a square represents
2.5 miles. If they start their hike at (Ϫ3, 1), how far must they
hike to reach the campsite?
© Glencoe/McGraw-Hill 418 Advanced Mathematical Concepts
10-1 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Mathematics and History: Hypatia
Hypatia (A.D. 370 –415) is the earliest woman mathematician whose
life is well documented. Born in Alexandria, Egypt, she was widely
known for her keen intellect and extraordinary mathematical ability.
Students from Europe, Asia, and Africa flocked to the university at
Alexandria to attend her lectures on mathematics, astronomy,
philosophy, and mechanics.
Hypatia wrote several major treatises in mathematics. Perhaps the
most significant of these was her commentary on the Arithmetica of
Diophantus, a mathematician who lived and worked in Alexandria
in the third century. In her commentary, Hypatia offered several obser-
vations about the Arithmetica’s Diophantine problems—
problems for which one was required to find only the rational
solutions. It is believed that many of these observations were
subsequently incorporated into the original manuscript of the
Arithmetica.
In modern mathematics, the solutions of a Diophantine equation
are restricted to integers. In the exercises, you will explore some
questions involving simple Diophantine equations.
For each equation, find three solutions that consist of an ordered
pair of integers.
1. 2x Ϫ y ϭ 7 2. x ϩ 3y ϭ 5
3. 6x Ϫ 5y ϭ Ϫ8 4. Ϫ11x Ϫ 4y ϭ 6
5. Refer to your answers to Exercises 1–4. Suppose that the integer
pair (x1, y1) is a solution of Ax Ϫ By ϭ C. Describe how to f ind
other integer pairs that are solutions of the equation.
6. Explain why the equation 3x ϩ 6y ϭ 7 has no solutions that are
integer pairs.
7. True or false: Any line on the coordinate plane must pass through
at least one point whose coordinates are integers. Explain.
© Glencoe/McGraw-Hill 419 Advanced Mathematical Concepts
10-2 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Circles
The standard form of the equation of a circle with radius r
and center at (h, k) is (x Ϫ h)2 ϩ ( y Ϫ k)2 ϭ r2.
Example 1 Write the standard form of the equation of the
circle that is tangent to the x-axis and has its
center at (Ϫ4, 3). Then graph the equation.
Since the circle is tangent to the x-axis, the
distance from the center to the x-axis is the radius.
The center is 3 units above the x-axis. Therefore,
the radius is 3.
(x Ϫ h)2 ϩ ( y Ϫ k)2 ϭ r2 Standard form
[x Ϫ (Ϫ4)]2 ϩ ( y Ϫ 3)2 ϭ 32 (h, k) ϭ (Ϫ4, 3) and r ϭ 3
(x ϩ 4)2 ϩ ( y Ϫ 3)2 ϭ 9
Example 2 Write the standard form of the equation of the
circle that passes through the points at (1, Ϫ1),
(5, 3), and (Ϫ3, 3). Then identify the center and
radius of the circle.
Substitute each ordered pair (x, y) in the general
form x2 ϩ y2 ϩ Dx ϩ Ey ϩ F ϭ 0 to create a system
of equations.
(1)2 ϩ (Ϫ1)2 ϩ D(1) ϩ E(Ϫ1) ϩ F ϭ 0 (x, y) ϭ (1, Ϫ1)
(5)2 ϩ (3)2 ϩ D(5) ϩ E(3) ϩ F ϭ 0 (x, y) ϭ (5, 3)
(Ϫ3)2 ϩ (3)2 ϩ D(Ϫ3) ϩ E(3) ϩ F ϭ 0 (x, y) ϭ (Ϫ3, 3)
Simplify the system of equations.
DϪEϩFϩ2ϭ0
5D ϩ 3E ϩ F ϩ 34 ϭ 0
Ϫ3D ϩ 3E ϩ F ϩ 18 ϭ 0
The solution to the system is D ϭ Ϫ2, E ϭ Ϫ6, and F ϭ Ϫ6.
The general form of the equation of the circle is
x2 ϩ y2 Ϫ 2x Ϫ 6y Ϫ 6 ϭ 0.
x2 ϩ y2 Ϫ 2x Ϫ 6y Ϫ 6 ϭ 0
(x2 Ϫ 2x ϩ ?) ϩ ( y2 Ϫ 6y ϩ ?) ϭ 6 Group to form perfect square trinomials.
(x2 Ϫ 2x ϩ 1) ϩ ( y2 Ϫ 6y ϩ 9) ϭ 6 ϩ 1 ϩ 9 Complete the square.
(x Ϫ 1)2 ϩ ( y Ϫ 3)2 ϭ 16 Factor the trinomials.
After completing the square, the standard form of the
circle is (x Ϫ 1)2 ϩ ( y Ϫ 3)2 ϭ 16. Its center is at (1, 3), and
its radius is 4.
© Glencoe/McGraw-Hill 420 Advanced Mathematical Concepts
10-2 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Circles
Write the standard form of the equation of each circle described.
Then graph the equation.
1. center at (3, 3) tangent to the x-axis 2. center at (2, Ϫ1), radius 4
Write the standard form of each equation. Then graph the
equation.
3. x2 ϩ y2 Ϫ 8x Ϫ 6 y ϩ 21 ϭ 0 4. 4 x2 ϩ 4 y2 ϩ 16x Ϫ 8y Ϫ 5 ϭ 0
Write the standard form of the equation of the circle that passes
through the points with the given coordinates. Then identify the
center and radius.
5. (Ϫ3, Ϫ2), (Ϫ2, Ϫ3), (Ϫ4, Ϫ3) 6. (0, Ϫ1), (2, Ϫ3), (4, Ϫ1)
7. Geometry A square inscribed in a circle and centered at the
origin has points at (2, 2), (Ϫ2, 2), (2, Ϫ2) and (Ϫ2, Ϫ2). What
is the equation of the circle that circumscribes the square?
© Glencoe/McGraw-Hill 421 Advanced Mathematical Concepts
10-2 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Spheres
The set of all points in three-dimensional space
that are a fixed distance r (the radius), from a
fixed point C (the center), is called a sphere. The
equation below is an algebraic representation of
the sphere shown at the right.
(x – h) 2 ϩ ( y – k) 2 ϩ (z – l ) 2 ϭ r2
A line segment containing the center of a sphere
and having its endpoints on the sphere is called a
diameter of the sphere. The endpoints of a
diameter are called poles of the sphere. A great
circle of a sphere is the intersection of the sphere
and a plane containing the center of the sphere.
1. If x 2 ϩ y2 – 4y ϩ z2 ϩ 2 z – 4 ϭ 0 is an equation of a
sphere and (1, 4, –3) is one pole of the sphere, find the
coordinates of the opposite pole.
2. a. On the coordinate system at the right, sketch the
sphere described by the equation x 2 ϩ y 2 ϩ z2 ϭ 9.
b. Is P(2, –2, –2) inside, outside, or on the sphere?
c. Describe a way to tell if a point with coordinates
P(a, b, c) is inside, outside, or on the sphere with
equation x 2 ϩ y 2 ϩ z2 ϭ r2.
3. If x 2 ϩ y 2 ϩ z2 – 4x ϩ 6y – 2 z – 2 ϭ 0 is an equation of
a sphere, find the circumference of a great circle, and the
surface area and volume of the sphere.
4. The equation x2 ϩ y2 ϭ 4 represents a set of points in
three-dimensional space. Describe that set of points in
your own words. Illustrate with a sketch on the
coordinate system at the right.
© Glencoe/McGraw-Hill 422 Advanced Mathematical Concepts
10-3 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Ellipses
The standard form of the equation of an ellipse is
ᎏ(x Ϫaᎏ2h)2 ϩ ᎏ( y Ϫbᎏ2k)2 ϭ 1 when the major axis is horizontal.
In this case, a2 is in the denominator of the x term. The
standard form is ᎏ( y Ϫaᎏ2k)2 ϩ ᎏ(x Ϫbᎏ2h)2 ϭ 1 when the major
axis is vertical. In this case, a2 is in the denominator of
the y term. In both cases, c2 ϭ a2 Ϫ b2.
Example Find the coordinates of the center, the foci,
and the vertices of the ellipse with the equation
4x2 ϩ 9y2 ϩ 24x Ϫ 36y ϩ 36 ϭ 0. Then graph the
equation.
First write the equation in standard form.
4x2 ϩ 9y2 ϩ 24x Ϫ 36y ϩ 36 ϭ 0 GCF of x terms is 4;
4(x2 ϩ 6x ϩ ?) ϩ 9( y2 Ϫ 4y ϩ ?) ϭ Ϫ36 ϩ ? ϩ ? GCF of y terms is 9.
4(x2 ϩ 6x ϩ 9) ϩ 9( y2 Ϫ 4y ϩ 4) ϭ Ϫ36 ϩ 4(9) ϩ 9(4) Complete the square.
4(x ϩ 3)2 ϩ 9( y Ϫ 2)2 ϭ 36 Factor.
ᎏ(x ϩ9ᎏ3)2 ϩ ᎏ( y Ϫ4ᎏ2)2 ϭ 1 Divide each side by 36.
Now determine the values of a, b, c, h, and k. In all
ellipses, a2 Ͼ b2. Therefore, a2 ϭ 9 and b2 ϭ 4. Since a2
is the denominator of the x term, the major axis is
parallel to the x-axis.
a ϭ 3 b ϭ 2 c ϭ ͙ෆa2ෆϪෆෆb2ෆ or ͙ෆ5 h ϭ Ϫ3 k ϭ 2
center: (Ϫ3, 2) (h, k)
foci: (Ϫ3 Ϯ ͙ෆ5, 2) (h Ϯ c, k)
major axis vertices: (h Ϯ a, k)
(0, 2) and (Ϫ6, 2)
minor axis vertices: (h, k Ϯ b)
(Ϫ3, 4) and (Ϫ3, 0)
Graph these ordered pairs. Then complete the ellipse.
© Glencoe/McGraw-Hill 423 Advanced Mathematical Concepts
10-3 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Ellipses
Write the equation of each ellipse in standard form. Then find the
coordinates of its foci.
1. 2.
For the equation of each ellipse, find the coordinates of the center,
foci, and vertices. Then graph the equation.
3. 4 x2 ϩ 9y2 Ϫ 8x Ϫ 36y ϩ 4 ϭ 0 4. 25x2 ϩ 9 y2 Ϫ 50x Ϫ 90 y ϩ 25 ϭ 0
Write the equation of the ellipse that meets each set of conditions.
5. The center is at (1, 3), the major axis is 6. The foci are at (Ϫ2, 1) and
parallel to the y-axis, and one vertex is (Ϫ2, Ϫ7), and a ϭ 5.
at (1, 8), and b ϭ 3.
7. Construction A semi elliptical arch is used to design a
headboard for a bed frame. The headboard will have a height
of 2 feet at the center and a width of 5 feet at the base. Where
should the craftsman place the foci in order to sketch the arch?
© Glencoe/McGraw-Hill 424 Advanced Mathematical Concepts
10-3 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Superellipses
The circle and the ellipse are members of an interesting family of
curves that were first studied by the French physicist and
mathematician Gabriel Lame′ (1795-1870). The general equation for
the family is
Έ Έ Έ Έᎏax n ϩ ᎏby n ϭ 1, with a 0, b 0, and n Ͼ 0.
For even values of n greater than 2, the curves are called
superellipses.
1. Consider two curves that are not
superellipses. Graph each equation on the
grid at the right. State the type of curve
produced each time.
ᎏx 2ϩ ᎏy 2ϭ
2Έ Έ Έ Έa.2 1
x 2ϩ y 2ϭ1
ᎏ3Έ Έ Έ Έb.ᎏ2
2. In each of the following cases you are given
values of a, b, and n to use in the general
equation. Write the resulting equation.
Then graph. Sketch each graph on the grid
at the right.
a. a ϭ 2, b ϭ 3, n ϭ 4
b. a ϭ 2, b ϭ 3, n ϭ 6
c. a ϭ 2, b ϭ 3, n ϭ 8
Έ Έ Έ Έ3. What shape will the graph of ᎏ2x n ϩ ᎏ3y n ϭ 1
approximate for greater and greater even,
whole-number values of n?
© Glencoe/McGraw-Hill 425 Advanced Mathematical Concepts
10-4 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Hyperbolas
The standard form of the equation of a hyperbola is
ᎏ(x Ϫaᎏ2h)2 Ϫ ᎏ( y Ϫbᎏ2k)2 ϭ 1 when the transverse axis is horizontal, and
ᎏ( y Ϫaᎏ2k)2 Ϫ ᎏ(x Ϫbᎏ2h)2 ϭ 1 when the transverse axis is vertical. In both
cases, b2 ϭ c2 Ϫ a2.
Example Find the coordinates of the center, foci, and vertices,
and the equations of the asymptotes of the graph of
25x2 Ϫ 9y2 ϩ 100x Ϫ 54y Ϫ 206 ϭ 0. Then graph the equation.
Write the equation in standard form.
25x2 Ϫ 9y2 ϩ 100x Ϫ 54y Ϫ 206 ϭ 0 GCF of x terms is 25;
GCF of y terms is 9.
25(x2 ϩ 4x ϩ ?) Ϫ 9( y2 ϩ 6y ϩ ?) ϭ 206 ϩ ? ϩ ?
25(x2 ϩ 4x ϩ 4) Ϫ 9( y2 ϩ 6y ϩ 9) ϭ 206 ϩ 25(4) ϩ (Ϫ9)(9) Complete
the square.
25(x ϩ 2)2 Ϫ 9( y ϩ 3)2 ϭ 225 Factor .
ᎏ(x +9ᎏ2)2 Ϫ ᎏ(y ϩ2ᎏ53)2 ϭ 1 Divide each side by 225.
From the equation, h ϭ Ϫ2, k ϭ Ϫ3,
a ϭ 3, b ϭ 5, and c ϭ ͙ෆ34ෆ. The
center is at (Ϫ2, Ϫ3).
Since the x terms are in the first
expression, the hyperbola has a
horizontal transverse axis.
The vertices are at (h Ϯ a, k) or
(1, Ϫ3) and (Ϫ5, Ϫ3).
The foci are at (h Ϯ c, k) or
(Ϫ2 Ϯ ͙ෆ34ෆ, Ϫ3).
The equations of the asymptotes are
y Ϫ k ϭ Ϯᎏabᎏ(x Ϫ h) or y ϩ 3 ϭ Ϯᎏ53ᎏ(x ϩ 2).
Graph the center, the vertices, and the rectangle guide,
which is 2a units by 2b units. Next graph the
asymptotes. Then sketch the hyperbola.
© Glencoe/McGraw-Hill 426 Advanced Mathematical Concepts
10-4 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Hyperbolas
For each equation, find the coordinates of the center, foci, and vertices, and
the equations of the asymptotes of its graph. Then graph the equation.
1. x2 Ϫ 4y2 Ϫ 4 x ϩ 24y Ϫ 36 ϭ 0 2. y2 Ϫ 4 x2 ϩ 8 x ϭ 20
Write the equation of each hyperbola. 4.
3.
5. Write an equation of the hyperbola for which the length of the
transverse axis is 8 units, and the foci are at (6, 0) and (Ϫ4, 0).
6. Environmental Noise Two neighbors who live one mile apart
hear an explosion while they are talking on the telephone. One
neighbor hears the explosion two seconds before the other. If
sound travels at 1100 feet per second, determine the equation of
the hyperbola on which the explosion was located.
© Glencoe/McGraw-Hill 427 Advanced Mathematical Concepts
10-4 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Moving Foci
Recall that the equation of a hyperbola with center
at the origin and horizontal transverse axis has the
equation x2 – y2 = 1. The foci are at (–c, 0) and
ᎏa2 ᎏb2
(c, 0), where c2 ϭ a2 ϩ b2, the vertices are at (–a, 0)
and (a, 0), and the asymptotes have equations
y ϭ ± ᎏab x. Such a hyperbola is shown at the right.
What happens to the shape of the graph as c grows
very large or very small?
Refer to the hyperbola described above.
1. Write a convincing argument to show that as c approaches 0, the
foci, the vertices, and the center of the hyperbola become the
same point.
2. Use a graphing calculator or computer to graph x 2 – y 2 ϭ 1,
x 2 – y 2 ϭ 0.1, and x 2 – y 2 ϭ 0.01. (Such hyperbolas correspond
to smaller and smaller values of c.) Describe the changes in the
graphs. What shape do the graphs approach as c approaches 0?
3. Suppose a is held fixed and c approaches infinity. How does the
graph of the hyperbola change?
4. Suppose b is held fixed and c approaches infinity. How does the
graph of the hyperbola change?
© Glencoe/McGraw-Hill 428 Advanced Mathematical Concepts
10-5 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Parabolas
The standard form of the equation of the parabola is ( y Ϫ k)2 ϭ 4p(x Ϫ h)
when the parabola opens to the right. When p is negative, the parabola
opens to the left. The standard form is (x Ϫ h)2 ϭ 4p( y Ϫ k) when the
parabola opens upward. When p is negative, the parabola opens downward.
Example 1 Given the equation x2 ϭ 12y ϩ 60, find the coordinates
of the focus and the vertex and the equations of the
directrix and the axis of symmetry. Then graph the
equation of the parabola.
First write the equation in the form (x Ϫ h)2 ϭ 4p( y Ϫ k).
x2 ϭ 12y ϩ 60 Factor.
x2 ϭ 12( y ϩ 5) 4p ϭ 12, so p ϭ 3.
(x Ϫ 0)2 ϭ 4(3)( y ϩ 5)
In this form, we can see that h ϭ 0, k ϭ Ϫ5, and p ϭ 3.
Vertex: (0, Ϫ5) (h, k) Focus: (0, Ϫ2) (h, k ϩ p)
xϭh
Directrix: y ϭ Ϫ8 y ϭ k Ϫ p Axis of Symmetry: x ϭ 0
The axis of symmetry is the y-axis.
Since p is positive, the parabola opens
upward. Graph the directrix, the vertex,
and the focus. To determine the shape of
the parabola, graph several other
ordered pairs that satisfy the equation
and connect them with a smooth curve.
Example 2 Write the equation y2 ϩ 6y ϩ 8x ϩ 25 ϭ 0 in standard
form. Find the coordinates of the focus and the
vertex, and the equations of the directrix and the axis of
symmetry. Then graph the parabola.
y2 ϩ 6y ϩ 8x ϩ 25 ϭ 0 Isolate the x terms and the y terms.
y2 ϩ 6y ϭ Ϫ8x Ϫ 25
Complete the square.
y2 ϩ 6y ϩ ? ϭ Ϫ8x Ϫ 25 ϩ ? Simplify and factor.
y2 ϩ 6y ϩ 9 ϭ Ϫ8x Ϫ 25 ϩ 9
( y ϩ 3)2 ϭ Ϫ8(x ϩ 2)
From the standard form, we can see that h ϭ Ϫ2
and k ϭ Ϫ3. Since 4p ϭ Ϫ8, p ϭ Ϫ2. Since y is
squared, the directrix is parallel to the y-axis. The
axis of symmetry is the x-axis. Since p is negative,
the parabola opens to the left.
Vertex: (Ϫ2, Ϫ3) (h, k)
Focus: (Ϫ4, Ϫ3) (h ϩ p, k)
Directrix: x ϭ 0 xϭhϪp
Axis of Symmetry: y ϭ Ϫ3 yϭk
© Glencoe/McGraw-Hill 429 Advanced Mathematical Concepts
10-5 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Parabolas
For the equation of each parabola, find the coordinates of the
vertex and focus, and the equations of the directrix and axis of
symmetry. Then graph the equation.
1. x2 Ϫ 2 x Ϫ 8 y ϩ 17 ϭ 0 2. y2 ϩ 6 y ϩ 9 ϭ 12 Ϫ 12x
Write the equation of the parabola that meets each set of
conditions. Then graph the equation.
3. The vertex is at (Ϫ2, 4) and 4. The focus is at (2, 1), and the
the focus is at (Ϫ2, 3). equation of the directrix is x ϭ Ϫ2.
5. Satellite Dish Suppose the receiver in a parabolic dish antenna
is 2 feet from the vertex and is located at the focus. Assume that
the vertex is at the origin and that the dish is pointed upward.
Find an equation that models a cross section
of the dish.
© Glencoe/McGraw-Hill 430 Advanced Mathematical Concepts
10-5 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Tilted Parabolas
The diagram at the right shows a fixed point F(1, 1)
and a line d whose equation is y ϭ –x Ϫ 2. If P(x, y)
satisfies the condition that PD ϭ PF, then P is on a
parabola. Our objective is to find an equation for the
tilted parabola; which is the locus of all points that are
the same distance from (1,1) and the line y ϭ –x Ϫ 2.
To do this, first find an equation for the line m through
P(x, y) and perpendicular to line d at D(a, b). Using
this equation and the equation for line d, find the
coordinates (a, b) of point D in terms of x and y. Then
use ( PD) 2 ϭ ( PF ) 2 to find an equation for the parabola.
Refer to the discussion above.
1. Find an equation for line m.
2. Use the equations for lines m and d to show that the coordinates
of point D are D(a, b) ϭ D ᎏx Ϫ 2yᎏϪ 2 , ᎏy Ϫ 2xᎏϪ 2 .
3. Use the coordinates of F, P, and D, along with ( PD) 2 ϭ ( PF ) 2 to
find an equation of the parabola with focus F and directrix d.
4. a. Every parabola has an axis of symmetry. Find an equation for
the axis of symmetry of the parabola described above. Justify
your answer.
b. Use your answer from part a to find the coordinates of the
vertex of the parabola. Justify your answer.
© Glencoe/McGraw-Hill 431 Advanced Mathematical Concepts
10-6 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Rectangular and Parametric Forms of Conic Sections
Use the table to identify a conic section given its equation in
general form.
conic Ax2 ϩ Bxy ϩ Cy2 ϩ Dx ϩ Ey ϩ F ϭ 0
circle AϭC
parabola Either A or C is zero.
ellipse A and C have the same sign and A C.
hyperbola A and C have opposite signs.
Example 1 Identify the conic section represented by the
equation 5x2 ϩ 4y2 Ϫ 10x Ϫ 8y ϩ 18 ϭ 0.
A ϭ 5 and C ϭ 4. Since A and C have the same
signs and are not equal, the conic is an ellipse.
Example 2 Find the rectangular equation of the curve whose
parametric equations are x ϭ 2t and y ϭ 4t2 ϩ 4t Ϫ 1.
Then identify the conic section represented by
the equation.
First, solve the equation x ϭ 2t Then substitute ᎏ2xᎏ for t in the
for t. equation y ϭ 4t2 ϩ 4t Ϫ 1.
x ϭ 2t y ϭ 4t2 ϩ 4t Ϫ 1 t ϭ ᎏx2ᎏ
ᎏ2xᎏ ϭ t
y ϭ 4 ᎏ2xᎏ 2 ϩ 4ᎏ2xᎏ Ϫ 1
y ϭ x2 ϩ 2x Ϫ 1
Since C ϭ 0, the equation y ϭ x2 ϩ 2x Ϫ 1 is the
equation of a parabola.
Example 3 Find the rectangular equation of the curve whose
parametric equations are x ϭ 3 cos t and y ϭ 5 sin t,
where 0 Յ t Յ 2. Then graph the equation using
arrows to indicate orientation.
Solve the first equation for cos t This is the equation of an ellipse with
the center at (0, 0). As t increases
and the second equation for sin t. from 0 to 2, the curve is traced in a
cos t ϭ ᎏ3xᎏ and sin t ϭ ᎏy5ᎏ counterclockwise motion.
Use the trigonometric identity
cos2 t ϩ sin2 t ϭ 1 to eliminate t.
cos2 t ϩ sin2 t ϭ 1 Substitution
ᎏ3xᎏ 2 ϩ ᎏ5yᎏ 2 ϭ 1
ᎏx9ᎏ2 ϩ ᎏ2y5ᎏ2 ϭ 1
© Glencoe/McGraw-Hill 432 Advanced Mathematical Concepts
10-6 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Rectangular and Parametric Forms of Conic Sections
Identify the conic section represented by each equation. Then
write the equation in standard form and graph the equation.
1. x2 Ϫ 4y ϩ 4 ϭ 0 2. x2 ϩ y2 Ϫ 6x Ϫ 6 y Ϫ 18 ϭ 0
3. 4 x2 Ϫ y2 Ϫ 8x ϩ 6 y ϭ 9 4. 9x2 ϩ 5y2 ϩ 18x ϭ 36
Find the rectangular equation of the curve whose parametric
equations are given. Then graph the equation using arrows to
indicate orientation.
5. x ϭ 3 cos t, y ϭ 3 sin t, 0 Յ t Յ 2 6. x ϭ Ϫ4 cos t, y ϭ 5 sin t, 0 Յ t Յ 2
© Glencoe/McGraw-Hill 433 Advanced Mathematical Concepts
10-6 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Polar Graphs of Conics
A conic is the locus of all points such that the ratio e of the y
Ox
distance from a fixed point F and a fixed line d is constant.
e > 1: hyperbola
FP ϭ e
ᎏDP y
To find the polar equation of the conic, use a polar
coordinate system with the origin at the focus.
Since FP ϭ r and DP ϭ p ϩ r cos , ᎏp ϩ rrᎏcos ϭ e.
r ϭ ᎏ1 – eepᎏcos
Now solve for r.
You can classify a conic section by its eccentricity.
e ϭ 1: parabola 0 < e < 1: ellipse
e ϭ 0: circle
yy
Ox Ox Ox
Graph each relation and identify the conic.
1. r ϭ ᎏ1 – 4cᎏos 2. r ϭ ᎏ2 – 4cᎏos
3. r ϭ ᎏ2 ϩ 4sᎏin 4. r ϭ ᎏ1 ϩ 24ᎏsin
© Glencoe/McGraw-Hill 434 Advanced Mathematical Concepts
10-7 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Transformation of Conics
Translations are often written in the form T(h, k). To find the
equation of a rotated conic, replace x with x′ cos ϩ y′ sin and
y with Ϫx′ sin ϩ y′ cos .
Example Identify the graph of each equation. Write an
equation of the translated or rotated graph in
general form.
a. 4 x2 ϩ y2 ϭ 12 for T(Ϫ2, 3)
The graph of this equation is an ellipse. To
write the equation of 4x2 ϩ y2 ϭ 12 for T(Ϫ2, 3),
let h ϭ Ϫ2 and k ϭ 3. Then replace x with
x Ϫ h and y with y Ϫ k.
x2 ⇒ (x Ϫ (Ϫ2))2 or (x ϩ 2)2
y2 ⇒ ( y Ϫ 3)2
Thus, the translated equation is 4(x ϩ 2)2 ϩ ( y Ϫ 3)2 ϭ 12.
Write the equation in general form.
4(x ϩ 2)2 ϩ ( y Ϫ 3)2 ϭ 12 Expand the binomial.
4(x2 ϩ 4x ϩ 4) ϩ y2 Ϫ 6y ϩ 9 ϭ 12 Simplify.
Subtract 12 from both sides.
4x2 ϩ y2 ϩ 16x Ϫ 6y ϩ 25 ϭ 12
4x2 ϩ y2 ϩ 16x Ϫ 6y ϩ 13 ϭ 0
b. x2 Ϫ 4y ϭ 0, ϭ 45°
The graph of this equation is a parabola. Find
the expressions to replace x and y.
Replace x with x′ cos 45؇ ϩ y′ sin 45؇ or ᎏ͙2ᎏෆ2 x′ ϩ ᎏ͙2ᎏ2ෆ y′.
Replace y with Ϫx′ sin 45؇ ϩ y′ cos 45؇ or Ϫᎏ͙2ᎏ2ෆ x′ ϩ ᎏ͙2ᎏෆ2 y′.
x2 Ϫ 4y ϭ 0
ᎏ͙2ᎏෆ2 x′ ϩ ᎏ͙2ᎏ2ෆ y′ 2 Ϫ 4 Ϫᎏ͙2ᎏෆ2 x′ ϩ ᎏ͙2ᎏෆ2 y′ ϭ 0 Replace x and y.
΄ ΅ ᎏ12ᎏ(x′)2 ϩ x′y′ ϩ ᎏ21ᎏ( y′)2 Ϫ 4 Ϫᎏ͙2ᎏ2ෆ x′ ϩ ᎏ͙2ᎏෆ2 y′ ϭ 0 Expand the binomial.
Simplify.
ᎏ21ᎏ(x′)2 ϩ x′y′ ϩ ᎏ12ᎏ( y′)2 ϩ 2͙ෆ2x′ Ϫ 2͙ෆ2y′ ϭ 0
The equation of the parabola after the 45؇ rotation is
ᎏ21ᎏ(x′)2 ϩ x′y′ ϩ ᎏ21ᎏ( y′)2 ϩ 2͙ෆ2x′ Ϫ 2͙2ෆy′ ϭ 0
© Glencoe/McGraw-Hill 435 Advanced Mathematical Concepts
10-7 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Transformations of Conics
Identify the graph of each equation. Write an equation of the
translated or rotated graph in general form.
1. 2 x2 ϩ 5 y2 ϭ 9 for T(Ϫ2, 1) 2. 2 x2 Ϫ 4 x ϩ 3 Ϫ y ϭ 0 for T(1, Ϫ1)
3. xy ϭ 1, ϭ ᎏ4ᎏ 4. x2 Ϫ 4y ϭ 0, ϭ 90°
Identify the graph of each equation. Then find to the
nearest degree.
5. 2 x2 ϩ 2 y2 Ϫ 2 x ϭ 0 6. 3 x2 ϩ 8 xy ϩ 4 y2 Ϫ 7 ϭ 0
7. 16x2 Ϫ 24 xy ϩ 9 y2 Ϫ 30x Ϫ 40 y ϭ 0 8. 13x2 Ϫ 8xy ϩ 7y2 Ϫ 45 ϭ 0
9. Communications Suppose the orientation of a satellite
dish that monitors radio waves is modeled by the equation
4x2 ϩ 2xy ϩ 4 y2 ϩ ͙ෆ2 x Ϫ ͙ෆ2 y ϭ 0. What is the angle of
rotation of the satellite dish about the origin?
© Glencoe/McGraw-Hill 436 Advanced Mathematical Concepts
10-7 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Graphing with Addition of y-Coordinates
Equations of parabolas, ellipses, and hyperbolas
that are “tipped” with respect to the x- and y-axes
are more difficult to graph than the equations you
have been studying.
Often, however, you can use the graphs of two
simpler equations to graph a more complicated
equation. For example, the graph of the ellipse in
the diagram at the right is obtained by adding the
y-coordinate of each point on the circle and the
y-coordinate of the corresponding point of the line.
Graph each equation. State the type of curve for each graph.
1. y ϭ 6 Ϫ x Ϯ ͙ෆ4ෆϪෆxෆ2 2. y ϭ x Ϯ ͙ෆx
Use a separate sheet of graph paper to graph these equations. State the type of
curve for each graph.
3. y ϭ 2x Ϯ ͙ෆ7ෆϩෆ6xෆෆϪෆxෆ2 4. y ϭ Ϫ2x Ϯ ͙–ෆ2ෆxෆ
© Glencoe/McGraw-Hill 437 Advanced Mathematical Concepts
10-8 NAME _____________________________ DATE _______________ PERIOD ________
Study Guide
Systems of Second-Degree Equations and
Inequalities
To find the exact solution to a system of second-degree equations,
you must use algebra. Graph systems of inequalities involving
second-degree equations to find solutions for the inequality.
Example a. Solve the system of equations algebraically.
Round to the nearest tenth.
x2 ϩ 2y2 ϭ 9
3x2 Ϫ y2 ϭ 1
Since both equations contain a single term involving y,
you can solve the system as follows.
First, multiply each side of Then, add the equations.
the second equation by 2. 6x2 Ϫ 2y2 ϭ 2
2(3x2 Ϫ y2) ϭ 2(1) x2 ϩ 2y2 ϭ 9
6x2 Ϫ 2y2 ϭ 2
7x2 ϭ 11
x ϭ ϮΊᎏ171ᎏ
ΊNow find y by substituting Ϯᎏ17ᎏ1 for x in one
of the original equations.
2
Ίx2 ϩ 2y2 ϭ 9 → Ϯ ᎏ17ᎏ1 ϩ2y2 ϭ 9
11 ϩ 14y2 ϭ 63
y2 ϭ ᎏ27ᎏ6
y Ϸ Ϯ1.9
The solutions are (1.3, Ϯ1.9), and (Ϫ1.3, Ϯ1.9).
b. Graph the solutions for the system of
inequalities.
x2 ϩ 2y2 Ͻ 9
3x2 Ϫ y2 Ն 1
First graph x2 ϩ 2y2 Ͻ 9. The ellipse should be
dashed. Test a point either inside or outside
the ellipse to see if its coordinates satisfy the
inequality. Since (0, 0) satisfies the inequality,
shade the interior of the ellipse.
Then graph 3x2 Ϫ y2 Ն 1. The hyperbola
should be a solid curve. Test a point inside
the branches of the hyperbola or outside its
branches. Since (0, 0) does not satisfy the
inequality, shade the regions inside the
branches. The intersection of the two graphs
represents the solution of the system.
© Glencoe/McGraw-Hill 438 Advanced Mathematical Concepts
10-8 NAME _____________________________ DATE _______________ PERIOD ________
Practice
Systems of Second-Degree Equations and Inequalities
Solve each system of equations algebraically. Round to the
nearest tenth. Check the solutions by graphing each system.
1. 2x Ϫ y ϭ 8 2. x2 Ϫ y2 ϭ 4
x2 ϩ y2 ϭ 9 yϭ1
3. xy ϭ 4 4. x2 ϩ y2 ϭ 4
x2 ϭ y2 ϩ 1 4 x2 ϩ 9 y2 ϭ 36
Graph each system of inequalities. 6. (x Ϫ 1)2 ϩ ( y Ϫ 2)2 Ͻ 9
4( y ϩ 1)2 ϩ x2 Յ 16
5. 3 Ն ( y Ϫ 1)2 ϩ 2x
y Ն Ϫ3x ϩ 1
7. Sales Vincent’s Pizzeria reduced prices for large specialty
pizzas by $5 for 1 week in March. In the previous week, sales for
large specialty pizzas totaled $400. During the sale week, the
number of large pizzas sold increased by 20 and total sales
amounted to $600. Write a system of second-degree equations to
model this situation. Find the regular price and the sale price
of large specialty pizzas.
© Glencoe/McGraw-Hill 439 Advanced Mathematical Concepts
10-8 NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Intersections of Circles
Many interesting problems involving circles can be
solved by using a system of equations. Consider the
following problem.
Find an equation for the straight line that contains
the two points of intersection of two intersecting
circles whose equations are given.
You may be surprised to find that if the given circles
intersect in two points, then the difference of their
equations is the equation of the line containing the
intersection points.
1. Circle A has equation x 2 ϩ y2 ϭ 1 and circle B has
equation (x – 3)2 ϩ y2 ϭ 1. Use a sketch to show that the
circles do not intersect. Use an algebraic argument to
show that circles A and B do not intersect.
2. Circle A has equation (x – 2)2 ϩ ( y ϩ 1)2 ϭ 16 and
circle B has equation (x ϩ 3)2 ϩ y2 ϭ 9. Use a sketch to
show that the circles meet in two points. Then find an
equation in standard form for the line containing the
points of intersection.
3. Without graphing the equations, decide if the circles with
equations (x – 2)2 ϩ ( y – 2)2 ϭ 8 and (x – 3)2 ϩ ( y – 4)2 ϭ 4 are
tangent. Justify your answer.
© Glencoe/McGraw-Hill 440 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 1A
Write the letter for the correct answer in the blank at the right of each problem.
Exercises 1–3 refer to the ellipse represented by 9x 2 ϩ 16y 2 Ϫ 18x ϩ 6 4y Ϫ 71 ϭ 0.
1. Find the coordinates of the center. 1. ________
A. (1, 2) B. (1, Ϫ2) C. (Ϫ1, 2) D. (Ϫ2, 1)
2. Find the coordinates of the foci. B. (1, Ϫ2 Ϯ ͙7ෆ) 2. ________
A. (1 Ϯ ͙7ෆ, Ϫ2)
C. (5, Ϫ2), (Ϫ3, Ϫ2) D. (1, 4), (1, Ϫ8)
3. Find the coordinates of the vertices. 3. ________
A. (1, 2), (1, Ϫ6), (4, Ϫ2), (Ϫ2, Ϫ2) B. (4, 2), (Ϫ2, 2), (1, 1), (1, Ϫ5)
C. (5, Ϫ2), (Ϫ3, Ϫ2), (1, 1), (1, Ϫ5) D. (5, Ϫ2), (Ϫ3, Ϫ2), (1, 2), (1, Ϫ6)
4. Write the standard form of the equation of the circle that passes 4. ________
through the points at (4, 5), (Ϫ2, 3), and (Ϫ4, Ϫ3).
A. (x Ϫ 5) 2 ϩ ( y ϩ 4)2 ϭ 49 B. (x Ϫ 3)2 ϩ ( y ϩ 2)2 ϭ 50
C. (x ϩ 4)2 ϩ ( y Ϫ 2)2 ϭ 36 D. (x Ϫ 2)2 ϩ ( y ϩ 2)2 ϭ 25
5. For 4x2 Ϫ 4xy ϩ y2 ϭ 4, find , the angle of rotation about the origin, 5. ________
to the nearest degree.
A. 27° B. 63° C. 333° D. 307°
6. Find the rectangular equation of the curve whose parametric 6. ________
equations are x ϭ 5 cos 2t and y ϭ Ϫsin 2t, 0° Յ t Յ 180°.
A. ᎏx5ᎏ2 ϩ y2 ϭ 1 B. ᎏx5ᎏ2 Ϫ y2 ϭ 1
C. ᎏ2xᎏ52 ϩ y2 ϭ 1 D. ᎏ(x Ϫᎏ2)2 ϩ ( y Ϫ 2) 2 ϭ 1
5
7. Find the distance between points at (m ϩ 4, n) and (m, n Ϫ 3). 7. ________
A. 3.5 B. 5 C. 1 D. 7
8. Write the standard form of the equation of the circle that is tangent 8. ________
to the line x ϭ Ϫ3 and has its center at (2, Ϫ7).
A. (x Ϫ 2 )2 ϩ ( y ϩ 7) 2 ϭ 25 B. (x Ϫ 2)2 ϩ ( y ϩ 7)2 ϭ 5
C. (x Ϫ 2) 2 ϩ ( y ϩ 7) 2 ϭ 16 D. (x ϩ 2)2 ϩ ( y Ϫ 7)2 ϭ 25
9. Find the coordinates of the point(s) of intersection for the graphs of 9. ________
x2 ϩ 2 y2 ϭ 33 and x2 ϩ y2 ϭ 2 x ϩ 19.
A. (5, 2), (Ϫ1, 4) B. (5, 4), (Ϫ1, 2)
C. (5, Ϯ2), (Ϫ1, Ϯ4) D. Graphs do not intersect.
10. Identify the conic section represented by 10. ________
9 y2 ϩ 4 x2 Ϫ 108y ϩ 24 x ϭ Ϫ144.
A. parabola B. hyperbola C. ellipse D. circle
11. Write the equation of the conic section y2 Ϫ x2 ϭ 5 after a rotation of 11. ________
45° about the origin.
A. xЈyЈ ϭ Ϫ2.5 B. xЈyЈ ϭ Ϫ5
C. ( yЈ)2 Ϫ (xЈ)2 ϭ 2.5 D. (xЈ)2 ϭ 2.5yЈ
12. Find parametric equations for the rectangular equation 12. ________
(x ϩ 2)2 ϭ 4( y Ϫ 1).
A. x ϭ t, y ϭ t 2 ϩ 2, Ϫ∞ Ͻ t Ͻ ∞
B. x ϭ t, y ϭ ᎏ41ᎏt 2 ϩ t ϩ 2, Ϫ∞ Ͻ t Ͻ ∞
C. x ϭ t, y ϭ ᎏ14ᎏt 2 Ϫ t ϩ 2, Ϫ∞ Ͻ t Ͻ ∞
D. x ϭ t, y ϭ 4t2 ϩ t ϩ 2, Ϫ∞ Ͻ t Ͻ ∞
© Glencoe/McGraw-Hill 441 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 1A (continued)
13. Which is the graph of 6x2 Ϫ 12x ϩ 6y2 ϩ 36y ϭ 36? 13. ________
A. B. C. D.
Exercises 14 and 15 refer to the hyperbola represented by
Ϫ2x2 ϩ y2 ϩ 4x ϩ 6y ϭ Ϫ3.
14. Write the equations of the asymptotes. 14. ________
B. y ϩ 3 ϭ Ϯᎏ12ᎏ(x Ϫ 1) 15. ________
A. y ϩ 3 ϭ Ϯ2(x Ϫ 1)
C. y ϩ 3 ϭ Ϯ͙ෆ2(x Ϫ 1) D. y ϩ 3 ϭ Ϯᎏ͙2ᎏ2ෆ (x Ϫ 1)
15. Find the coordinates of the foci.
A. (1 Ϯ ͙ෆ2, Ϫ3)B. (1 Ϯ ͙ෆ6, Ϫ3)C. (1, Ϫ3 Ϯ ͙ෆ2)D. (1, Ϫ3 Ϯ ͙ෆ6 )
16. Write the standard form of the equation of the hyperbola for which 16. ________
17. ________
the transverse axis is 4 units long and the coordinates of the foci
are (1, Ϫ4 Ϯ ͙ෆ7).
A. (x Ϫ 1)2 Ϫ ( y ϩ 4)2 ϭ 1 B. ( y ϩ 4)2 Ϫ (x Ϫ 1)2 ϭ 1
ᎏ3ᎏ ᎏ4ᎏ ᎏ4ᎏ ᎏ3ᎏ
( y ϩ 4)2 (x Ϫ 1)2 (x Ϫ 1) 2 ( y ϩ 4) 2
C. ᎏ3ᎏ Ϫ ᎏ4ᎏ ϭ 1 D. ᎏ4ᎏ Ϫ ᎏ3ᎏ ϭ 1
17. The graph at the right shows the solution set for
which system of inequalities?
A. xy Ն Ϫ6, 9(x ϩ 1)2 ϩ 4( y Ϫ 1)2 Յ 36
B. xy Յ Ϫ6, 4(x ϩ 1)2 ϩ 9( y Ϫ 1)2 Յ 36
C. xy Ն Ϫ6, 4(x ϩ 1)2 ϩ 9( y Ϫ 1)2 Յ 36
D. xy Յ Ϫ6, 9(x ϩ 1)2 ϩ 4( y Ϫ 1)2 Յ 36
18. Find the coordinates of the vertex and the equation
of the axis of symmetry for the parabola represented 18. ________
by x2 ϩ 4x Ϫ 6 y ϩ 10 ϭ 0.
A. (Ϫ2, 1), y ϭ 1 B. (1, Ϫ2), y ϭ Ϫ2
C. (Ϫ2, 1), x ϭ Ϫ2 D. (1, Ϫ2), x ϭ 1
19. Write the standard form of the equation of the parabola whose 19. ________
20. ________
directrix is x ϭ Ϫ1 and whose focus is at (5, Ϫ2).
A. ( y ϩ 2)2 ϭ 12(x ϩ 2) B. y Ϫ 2 ϭ 12(x ϩ 2)2
C. x ϩ 2 ϭ ᎏ11ᎏ2 ( y ϩ 2)2 D. x Ϫ 2 ϭ ᎏ11ᎏ2 ( y ϩ 2)2
20. Identify the graph of the equation 16x2 Ϫ 24xy ϩ 9y2 Ϫ 30x Ϫ 40y ϭ 0.
Then, find to the nearest degree.
A. hyperbola; Ϫ37° B. parabola; Ϫ37°
C. parabola; 45° D. circle; Ϫ74°
Bonus Find the coordinates of the points of intersection of the Bonus: ________
graphs of x2 Ϫ y 2 ϭ 3, xy ϭ 2, and y ϭ Ϫ2x ϩ 5.
A. (Ϯ2, Ϯ1) B. (1, Ϫ2)
C. (2, 1) D. Graphs do not intersect.
© Glencoe/McGraw-Hill 442 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 1B
Write the letter for the correct answer in the blank at the right of each problem.
Exercises 1-3 refer to the ellipse represented by x2 ϩ 25y2 Ϫ 6x Ϫ 100y ϩ 84 ϭ 0.
1. Find the coordinates of the center. 1. ________
A. (2, 3) B. (3, 2) C. (Ϫ3, Ϫ2) D. (Ϫ2, Ϫ3)
2. Find the coordinates of the foci. 2. ________
A. (3, 2 Ϯ ͙2ෆෆ6) B. (Ϫ2, 2), (8, 2) C. (3 Ϯ ͙2ෆෆ6, 2)D. (2 Ϯ ͙2ෆ6ෆ, 3)
3. Find the coordinates of the vertices. 3. ________
A. (8, 2), (Ϫ2, 2), (3, 3), (3, 1) B. (8, 2), (Ϫ2, 2), (3, 7), (3 , Ϫ3)
C. (4, 2), (2, 2), (3, 3), (3, 1) D. (4, 2), (2, 2), (3, 7), (3, Ϫ3)
4. Write 6x2 Ϫ 12x ϩ 6y2 ϩ 36y ϭ 36 in standard form. 4. ________
A. (x Ϫ 3)2 ϩ ( y Ϫ 1)2 ϭ 16 B. (x ϩ 1)2 ϩ ( y Ϫ 3)2 ϭ 16
C. (x Ϫ 1)2 ϩ ( y ϩ 3)2 ϭ 16 D. (x Ϫ 3)2 ϩ ( y ϩ 1)2 ϭ 16
5. For 2x2 ϩ 3xy ϩ y2 ϭ 1, find , the angle of rotation about the origin, 5. ________
to the nearest degree.
A. Ϫ9؇ B. 36؇ C. Ϫ36؇ D. 324؇
6. Find the rectangular equation of the curve whose parametric equations 6. ________
are x ϭ 3 cos t and y ϭ sin t, 0؇ Յ t Յ 360؇.
A. ᎏx9ᎏ2 ϩ y2 ϭ 1 B. ᎏx9ᎏ2 Ϫ y2 ϭ 1 C. y2 Ϫ ᎏx3ᎏ2 ϭ 1 D. y2 ϩ ᎏx3ᎏ2 ϭ 1
7. Find the distance between points at (Ϫ5, 2) and (7, Ϫ3). 7. ________
A. ͙ෆ5 B. 13 C. ͙2ෆ9ෆ D. ͙ෆ11ෆ9ෆ
8. Write the standard form of the equation of the circle that is tangent to 8. ________
the y-axis and has its center at (Ϫ3, 5).
A. (x ϩ 3)2 ϩ ( y Ϫ 5)2 ϭ 9 B. (x ϩ 3)2 ϩ ( y Ϫ 5)2 ϭ 25
C. (x ϩ 3)2 ϩ ( y Ϫ 5)2 ϭ 3 D. (x Ϫ 3)2 ϩ ( y ϩ 5)2 ϭ 9
9. Find the coordinates of the point(s) of intersection for the graphs of 9. ________
x2 ϩ y2 ϭ 4 and y ϭ 2x Ϫ 1.
A. (1.3, 1.5) B. (1.3, 1.5), (Ϫ0.5, Ϫ1.9)
C. (Ϯ1.3, Ϯ1.5), (Ϯ0.5, Ϯ1.9) D. Graphs do not intersect.
10. Identify the conic section represented by 3y2 Ϫ 3x2 ϩ 12y ϩ 18x ϭ 42. 10. ________
A. parabola B. hyperbola C. ellipse D. circle
11. Write the equation of the conic section y2 Ϫ x2 ϭ 2 after a rotation of 11. ________
45؇ about the origin.
A. x′y′ ϭ Ϫ1 B. x′y′ ϭ Ϫ2 C. ( y′)2 Ϫ (x′)2 ϭ 2 D. (x′)2 ϭ y′
12. Find parametric equations for the rectangular equation x2 ϩ y2 Ϫ 25 ϭ 0. 12. ________
A. x ϭ cos 5t, y ϭ sin 5t, 0 Յ t Յ 2 B. x ϭ cos t, y ϭ 5 sin t, 0 Յ t Յ 2
C. x ϭ 5 cos t, y ϭ sin t, 0 Յ t Յ 2 D. x ϭ 5 cos t, y ϭ 5 sin t, 0 Յ t Յ 2
© Glencoe/McGraw-Hill 443 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 1B (continued)
13. Which is the graph of x2 ϩ ( y ϩ 2)2 ϭ 16? 13. ________
A. B. C.
D.
Exercises 14 and 15 refer to the hyperbola represented by
36x2 Ϫ y2 Ϫ 4y ϭ Ϫ32.
14. Write the equations of the asymptotes. 14. ________
15. ________
A. y Ϫ 1 ϭ Ϯ6(x Ϫ 2) B. y ϭ Ϯ6x 16. ________
C. y ϩ 2 ϭ Ϯ6(x Ϫ 1) D. y ϩ 2 ϭ Ϯ6x 17. ________
15. Find the coordinates of the foci.
A. (1 Ϯ ͙ෆ37ෆ, Ϫ2) B. (Ϯ ͙ෆ3ෆ7, Ϫ2)
C. (6 Ϯ ͙ෆ3ෆ7, Ϫ2) D. (0, Ϫ2 Ϯ ͙3ෆ7ෆ)
16. Write the standard form of the equation of the hyperbola for which
a ϭ 2, the transverse axis is vertical, and the equations of the
asymptotes are y ϭ Ϯ2x.
A. ᎏx4ᎏ2 Ϫ y2 ϭ 1 B. y2 Ϫ ᎏx4ᎏ2 ϭ 1 C. x2 Ϫ ᎏy4ᎏ2 ϭ 1 D. ᎏy4ᎏ2 Ϫ x2 ϭ 1
17. The graph at the right shows the solution set for
which system of inequalities?
A. y Ϫ 1 Ն Ϫ(x Ϫ 1)2, 4(x Ϫ 1)2 ϩ 9( y ϩ 3)2 Յ 36
B. y Ϫ 1 Ն Ϫ(x Ϫ 1)2, 4(x Ϫ 1)2 ϩ 9( y ϩ 3)2 Ն 36
C. y Ϫ 1 Յ Ϫ(x Ϫ 1)2, 4(x Ϫ 1)2 ϩ 9( y ϩ 3)2 Յ 36
D. y Ϫ 1 Յ Ϫ(x Ϫ 1)2, 4(x Ϫ 1)2 ϩ 9( y ϩ 3)2 Ն 36
18. Find the coordinates of the vertex and the equation of the axis of 18. ________
symmetry for the parabola represented by x2 ϩ 2x ϩ 12y ϩ 37 ϭ 0.
A. (Ϫ1, Ϫ3), x ϭ Ϫ1 B. (Ϫ1, Ϫ6), x ϭ Ϫ1
C. (Ϫ1, Ϫ12), x ϭ Ϫ5 D. (3, 2), y ϭ Ϫ9
19. Write the standard form of the equation of the parabola whose 19. ________
directrix is y ϭ Ϫ4 and whose focus is at (2, 2).
A. ( y Ϫ 2)2 ϭ 12(x ϩ 2) B. y ϩ 1 ϭ 12(x Ϫ 2)2
C. (x ϩ 2)2 ϭ 12( y Ϫ 2) D. (x Ϫ 2)2 ϭ 12( y ϩ 1)
20. Identify the graph of the equation 4x2 Ϫ 5xy ϩ 16y2 Ϫ 32 ϭ 0. 20. ________
A. circle B. ellipse C. parabola D. hyperbola
Bonus Find the coordinates of the points of intersection of the Bonus: ________
graphs of x2 ϩ y2 ϭ 5, xy ϭ Ϫ2, and y ϭ Ϫ3x ϩ 1.
A. (Ϯ2, Ϯ1) B. (1, Ϫ2) C. (2, 1) D. (Ϯ1, Ϯ2)
© Glencoe/McGraw-Hill 444 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 1C
Write the letter for the correct answer in the blank at the right of each problem.
Exercises 1–3 refer to the ellipse represented by 4x2 ϩ 9 y 2 Ϫ 18y Ϫ 27 ϭ 0.
1. Find the coordinates of the center. 1. ________
A. (Ϫ1, 0) B. (0, Ϫ1) C. (1, 0) D. (0,1)
2. Find the coordinates of the foci. B. (͙ෆ5, 1), (Ϫ͙ෆ5, 1) 2. ________
A. (0, 1 Ϯ ͙ෆ5)
C. (͙5ෆ, 3), (Ϫ͙ෆ5, 3) D. (1, 5), (1, Ϫ5)
3. Find the coordinates of the vertices. 3. ________
A. (2, 1), (Ϫ2, 1) (0, 4), (0, Ϫ2) B. (3, 1), (Ϫ3, 1), (0, 3), (0, Ϫ1)
C. (3, 1), (Ϫ3, 1), (0, 4), (0, Ϫ2) D. (2, 1), (Ϫ2, 1), (0, 3), (0, Ϫ1)
4. Write the standard form of the equation of the circle that is tangent 4. ________
to the x-axis and has its center at (3, Ϫ2).
A. (x Ϫ 3)2 ϩ ( y ϩ 2)2 ϭ 4 B. (x ϩ 3)2 ϩ ( y Ϫ 2)2 ϭ 4
C. (x Ϫ 3)2 ϩ ( y ϩ 2)2 ϭ 2 D. (x ϩ 3)2 ϩ ( y Ϫ 2)2 ϭ 2
5. For 2x2 ϩ xy ϩ 2y2 ϭ 1, find , the angle of rotation about the origin, 5. ________
to the nearest degree.
A. 215° B. 150° C. 45° D. Ϫ30°
6. Find the rectangular equation of the curve whose parametric 6. ________
equations are x ϭ Ϫcos t and y ϭ sin t, 0° Յ t Յ 360°.
A. y2 Ϫ x2 ϭ 1 B. x2 ϩ y2 ϭ Ϫ1 C. x2 ϩ y2 ϭ 1 D. x2 Ϫ y2 ϭ 1
7. Find the distance between points at (Ϫ1, 6) and (5, Ϫ2). 7. ________
8. ________
A. ͙ෆ1ෆ4 B. 10 C. ͙3ෆ4ෆ D. 8
8. Write x2 ϩ 4 x ϩ y2 ϩ 2y ϭ 4 in standard form.
A. (x Ϫ 2)2 ϩ ( y Ϫ 1)2 ϭ 9 B. (x ϩ 2)2 ϩ ( y ϩ 1)2 ϭ 9
C. (x ϩ 2)2 ϩ ( y ϩ 1)2 ϭ 3 D. (x ϩ 2)2 Ϫ ( y Ϫ 1)2 ϭ 4
9. Find the coordinates of the point(s) of intersection for the graphs of 9. ________
x2 ϩ y2 ϭ 20 and y ϭ x Ϫ 2.
A. (Ϯ2, Ϯ4), (4, 2) B. (Ϯ2, Ϯ4), (Ϯ4, Ϯ2)
C. (2, 4), (Ϫ4, Ϫ2) D. (Ϫ2, Ϫ4), (4, 2)
10. Identify the conic section represented by x2 Ϫ y2 ϩ 12 y ϩ 18x ϭ 42. 10. ________
A. parabola B. hyperbola C. ellipse D. circle
11. Write the equation of the conic section x2 ϩ y2 ϭ 16 after a rotation 11. ________
of 45° about the origin.
A. (xЈ)2 ϩ ( yЈ)2 ϭ 16 B. (xЈ)2 Ϫ ( yЈ)2 ϭ 16
C. (xЈ)2 Ϫ 2 xЈyЈ ϩ ( yЈ)2 ϭ 16 D. (xЈ)2 ϩ 2xЈyЈ ϩ ( yЈ)2 ϭ 16
12. Find parametric equations for the rectangular equation x2 ϩ y2 ϭ 16. 12. ________
A. x ϭ cos 4t, y ϭ sin 4t, 0° Յ t Յ 360°
B. x ϭ cos 16t, y ϭ sin 16t, 0° Յ t Յ 360°
C. x ϭ 4 cos t, y ϭ 4 sin t, 0° Յ t Յ 360°
D. x ϭ 16 cos t, y ϭ 16 sin t, 0° Յ t Յ 360°
© Glencoe/McGraw-Hill 445 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 1C (continued)
13. Which is the graph of (x Ϫ 3)2 ϩ ( y Ϫ 2)2 ϭ 9? 13. ________
A. B. C.
D.
Exercises 14 and 15 refer to the hyperbola represented by
Ϫ16 y 2 Ϫ 5 4x ϩ 9x2 ϭ 63.
14. Write the equations of the asymptotes. 14. ________
15. ________
A. y Ϫ 3 ϭ Ϯ ᎏ34ᎏ x B. y Ϫ 3 ϭ Ϯ ᎏ43ᎏ x 16. ________
C. y ϭ Ϯ ᎏ43ᎏ(x Ϫ 3) D. y ϭ Ϯ ᎏ34ᎏ(x Ϫ 3)
17. ________
15. Find the coordinates of the foci.
A. (5, 0), (Ϫ5, 0) B. (0, 5), (0, Ϫ5)
C. (3, 5), (3, Ϫ5) D. (8, 0), (Ϫ2, 0)
16. Write the standard form of the equation of the hyperbola for
which a ϭ 5, b ϭ 6, the transverse axis is vertical, and the center
is at the origin.
A. y2 Ϫ x2 ϭ 1 B. x2 Ϫ y2 ϭ 1 C. x2 Ϫ y2 ϭ 1 D. y2 Ϫ x2 ϭ 1
ᎏ2ᎏ5 ᎏ3ᎏ6 ᎏ3ᎏ6 ᎏ25ᎏ ᎏ2ᎏ5 ᎏ3ᎏ6 ᎏ3ᎏ6 ᎏ2ᎏ5
17. The graph at the right shows the solution set for
which system of inequalities?
A. y2 Ϫ 4 x2 Յ 1, x2 ϩ y2 Յ 9
B. y2 Ϫ 4 x2 Ն 1, x2 ϩ y2 Յ 9
C. y2 Ϫ 4 x2 Յ 1, x2 ϩ y2 Յ 3
D. y2 Ϫ 4 x2 Ն 1, x2 ϩ y2 Յ 3
18. Find the coordinates of the vertex and the equation of the axis of 18. ________
symmetry for the parabola represented by y2 Ϫ 8x Ϫ 4 y ϩ 28 ϭ 0.
A. (3, 2), x ϭ 3 B. (3, 2), y ϭ 2
C. (2, 3), y ϭ 3 D. (2, 3), x ϭ 2
19. Write the standard form of the equation of the parabola whose 19. ________
directrix is x ϭ Ϫ2 and whose focus is at (2, 0).
A. ( y Ϫ 2)2 ϭ 8(x Ϫ 2) B. ( y Ϫ 2)2 ϭ 4(x Ϫ 2)
C. y2 ϭ 8 x D. x2 ϭ 8 y
20. Write the equation for the translation of the graph of 20. ________
y2 Ϫ 4 x Ϫ 12 ϭ 0 for T(Ϫ1, 1). B. ( y Ϫ 1)2 ϭ 4(x ϩ 4)
A. ( y Ϫ 1)2 ϭ 4(x ϩ 2)
C. ( y ϩ 1)2 ϭ 4(x Ϫ 2) D. ( y ϩ 1)2 ϭ 4(x Ϫ 4)
Bonus Find the coordinates of the points of intersection of the Bonus: ________
graphs of x2 Ϫ y2 ϭ 4, x2 ϩ y2 ϭ 4, and x Ϫ y ϭ 2.
A. (2, 0) B. (0, 2) C. (0, Ϫ2) D. (Ϫ2, 0)
© Glencoe/McGraw-Hill 446 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 2A
1. Find the distance between points at (m, n Ϫ 5) and 1. __________________
(m Ϫ 3, n ϩ 2). 2. __________________
2. Determine whether the quadrilateral ABCD with vertices 3. __________________
4. __________________
AϪ1, ᎏ21ᎏ , B ᎏ21ᎏ, Ϫᎏ21ᎏ , C1, Ϫᎏ23ᎏ , and D Ϫᎏ12ᎏ, Ϫ1 is a 5. __________________
parallelogram.
3. Write the standard form of the equation of the circle that
passes through the point at (2, Ϫ2) and has its
center at (Ϫ2, 3).
4. Write the standard form of the equation of the circle that
passes through the points at (1, 3), (7, 3), and (8, 2).
5. Write 2 x2 Ϫ 10x ϩ 2 y2 Ϫ 18 y Ϫ 1 ϭ 0 in standard form.
Then, graph the equation, labeling the center.
6. Write 3x2 ϩ 2 y2 ϩ 24 x Ϫ 4 y ϩ 26 ϭ 0 in standard 6. __________________
form. Then, graph the equation, labeling the
center, foci, and vertices.
7. Find the equation of the ellipse that has its major axis 7. __________________
parallel to the y-axis and its center at (4, Ϫ3), and that 8. __________________
passes through points at (1, Ϫ3) and (4, 2). 9. __________________
8. Find the equation of the equilateral hyperbola that has
its foci at (Ϫ2, Ϫ3 Ϫ 2͙ෆ3) and (Ϫ2, Ϫ3 ϩ 2͙ෆ3), and
whose conjugate axis is 6 units long.
9. Write Ϫ2 x2 ϩ 3y2 Ϫ 24x Ϫ 6 y Ϫ 93 ϭ 0 in standard
form. Find the equations of the asymptotes of the
graph. Then, graph the equation, labeling the center,
foci, and vertices.
© Glencoe/McGraw-Hill 447 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 2A (continued)
10. Write x ϭ y2 Ϫ 2 y Ϫ 5 in standard form. Find the equations 10. __________________
of the directrix and axis of symmetry. Then, graph the
equation, labeling the focus, vertex, and directrix.
11. Find the equation of the parabola that passes through the 11. __________________
12. __________________
point at 0, Ϫ ᎏ21ᎏ , has a vertical axis, and has a maximum
at (Ϫ2, 1).
12. Find the equation of the hyperbola that has eccentricity ᎏ59ᎏ
and foci at (4, 6) and (4, Ϫ12).
13. Identify the conic section represented by xy ϩ 3y ϩ 4x ϭ 0. 13. __________________
14. Find a rectangular equation for the curve whose 14. __________________
parametric equations are x ϭ Ϫᎏ21ᎏ cos 4 t ϩ 2,
y ϭ Ϫ2 sin 4t Ϫ 3, 0° Յ t Յ 90°.
15. Find parametric equations for the equation 15. __________________
ᎏ(x ϩ1ᎏ61)2 ϩ 2( y Ϫ 3)2 ϭ 1.
Identify the graph of each equation. Write an equation of the
translated or rotated graph in general form.
16. x2 ϩ 12 y ϭ Ϫ31 Ϫ 10x for T (3, Ϫ5) 16. ___________________________
17. x2 Ϫ xy ϩ 2 y2 ϭ 2, ϭ Ϫ30° 17. __________________
18. Identify the graph of 4 x2 ϩ 7xy Ϫ 5y2 ϭ Ϫ3. Then, find , 18. __________________
the angle of rotation about the origin, to the nearest degree.
19. Solve the system 4 x2 ϩ y 2 ϭ 32 Ϫ 4 y and x2 ϭ 7 ϩ y 19. __________________
algebraically. Round to the nearest tenth.
20. Graph the solutions for the system of inequalities. 20.
9x2 Ͼ y2
x2 Յ 100 Ϫ y2
Bonus Find the coordinates of the point(s) of Bonus: __________________
intersection of the graphs of 2x ϩ 1 ϭ y, Advanced Mathematical Concepts
x2 ϭ 10 Ϫ y2, and y Ϫ 4 x2 ϭ Ϫ1.
© Glencoe/McGraw-Hill 448
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 2B
1. Find the distance between points at (Ϫ2, Ϫ5) and (6, Ϫ1). 1. __________________
2. Determine whether the quadrilateral ABCD with vertices 2. __________________
A(Ϫ1, 2), B(2, 0), C(Ϫ2, Ϫ2), and D(Ϫ5, 0) is a parallelogram.
3. Write the standard form of the equation of the circle that is 3. __________________
tangent to x ϭ Ϫ3 and has its center at (1, Ϫ3).
4. Write the standard form of the equation of the circle that 4. __________________
passes through the points at (Ϫ6, 3), (Ϫ4, Ϫ1), and (Ϫ2, 5).
5. Write x2 ϩ y2 ϩ 6x Ϫ 14y Ϫ 42 ϭ 0 in standard form. Then, 5. __________________
graph the equation, labeling the center.
6. Write 4x2 ϩ 9y2 Ϫ 24x ϩ 18y ϩ 9 ϭ 0 in standard form. Then, 6. __________________
graph the equation, labeling the center, foci, and vertices.
7. Find the equation of the ellipse that has its foci at (2, 1) and 7. __________________
(2, Ϫ7) and b ϭ 2.
8. Find the equation of the hyperbola that has its foci at (0, Ϫ4) 8. __________________
and (10, Ϫ4), and whose conjugate axis is 6 units long.
9. Write 4x2 Ϫ y2 ϩ 24x ϩ 4y ϩ 28 ϭ 0 in standard form. Find 9. __________________
the equations of the asymptotes of the graph. Then, graph
the equation, labeling the center, foci, and vertices.
© Glencoe/McGraw-Hill 449 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 2B (continued)
10. Write Ϫ2x ϩ y2 Ϫ 2y ϩ 5 ϭ 0 in standard form. Find the 10. __________________
equations of the directrix and axis of symmetry. Then, graph
the equation, labeling the focus, vertex and directrix.
11. Find the equation of the parabola that passes through 11. __________________
the point at (Ϫ8, 15), has its vertex at (2, Ϫ5), and opens
to the left.
12. Find the equation of the ellipse that has its center at the 12. __________________
origin, eccentricity ᎏ2͙3ᎏ2ෆ , and a vertical major axis of 6 units.
13. Identify the conic section represented by x2 Ϫ 3xy ϩ y2 ϭ 5. 13. __________________
14. Find a rectangular equation for the curve whose parametric 14. __________________
equations are x ϭ cos 3t, y ϭ Ϫ2 sin 3t, 0؇ Յ t Յ 120؇.
15. Find parametric equations for the equation ᎏ1x6ᎏ2 ϩ ᎏ3yᎏ62 ϭ 1. 15. __________________
Identify the graph of each equation. Write an equation of the
translated or rotated graph in general form.
16. x2 Ϫ 2x Ϫ 2y Ϫ 9 ϭ 0 for T(Ϫ2, 3) 16. __________________
17. 2x2 ϩ 5y2 Ϫ 20 ϭ 0, ϭ 30؇ 17. __________________
18. Identify the graph of 3x2 Ϫ 8xy Ϫ 3y2 ϭ 3. Then, find , the 18. __________________
angle of rotation about the origin, to the nearest degree.
19. Solve the system 5x2 ϭ 10 Ϫ 2y2 and 3y2 ϭ 84 Ϫ 2x2 19. __________________
algebraically. Round to the nearest tenth.
20. Graph the solutions for the system of inequalities. 20.
(x ϩ 2)2 ϩ ( y Ϫ 1)2 Ͼ 9
ᎏ1xᎏ62 Ϫ y2 Յ 1
Bonus Find the coordinates of the point(s) of intersection Bonus: __________________
of the graphs of x ϩ y ϩ 1 ϭ 0, x2 ϩ y2 ϭ 5,
y ϭ Ϫ3x2 ϩ 1.
© Glencoe/McGraw-Hill 450 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 2C
1. Find the distance between points at (2, 1) and (5, Ϫ3). 1. __________________
2. Determine whether the quadrilateral ABCD with vertices 2. __________________
A(5, 3), B(7, 3), C(5, 1), and D(2, 1) is a parallelogram.
3. Write the standard form of the equation of the circle that 3. __________________
has its center at (Ϫ4, 3) and a radius of 5.
4. Write the standard form of the equation of the circle that 4. __________________
passes through the points at (Ϫ2, 2), (2, 2), and (2, Ϫ2).
5. Write x2 Ϫ 8x ϩ y2 ϩ 4 y ϩ 16 ϭ 0 in standard form. Then, 5. __________________
graph the equation, labeling the center.
6. Write 9x2 ϩ 54 x ϩ 4 y2 Ϫ 16 y ϩ 61 ϭ 0 in standard form. 6. __________________
Then, graph the equation, labeling the center, foci, and
vertices.
7. Find the equation of the ellipse that has its center at 7. __________________
(Ϫ1, 3), a horizontal major axis of 6 units, and a minor 8. __________________
axis of 2 units. 9. __________________
8. Find the equation of the hyperbola that has its center at
(Ϫ2, 4), a ϭ 3, b ϭ 5, and a vertical transverse axis.
9. Write x2 Ϫ 4 x Ϫ 4 y2 Ϫ 8 y Ϫ 16 ϭ 0 in standard
form. Find the equations of the asymptotes of
the graph. Then, graph the equation, labeling
the center, foci, and vertices.
© Glencoe/McGraw-Hill 451 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Test, Form 2C (continued)
10. Write y2 ϩ 4 y Ϫ 4 x ϩ 12 ϭ 0 in standard form. Find the 10. __________________
equations of the directrix and axis of symmetry. Then,
graph the equation, labeling the focus, vertex, and directrix.
11. Find the equation of the parabola that has its vertex at 11. __________________
(5, Ϫ1), and the focus at (5, Ϫ2). 12. __________________
12. Find the eccentricity of the ellipse (x Ϫ 2)2 ϩ ( y ϩ 3)2 ϭ 1.
ᎏ4ᎏ ᎏ8ᎏ
13. Identify the conic section represented by 13. __________________
x2 Ϫ 2y2 ϩ 16 y Ϫ 42 ϭ 0.
14. Find a rectangular equation for the curve whose parametric 14. __________________
equations are x ϭ 2 cos t, y ϭ Ϫ3 sin t, 0° Յ t Յ 360°.
15. Find parametric equations for the equation x2 ϩ y2 ϭ 8. 15. __________________
Identify the graph of each equation. Write an equation 16. __________________
of the translated or rotated graph in general form. 17. __________________
16. y2 ϩ 8 y ϩ 8x ϩ 32 ϭ 0 for T(1, Ϫ4)
17. 5x2 ϩ 4 y2 ϭ 20, ϭ 45°
18. Identify the graph of 4 x2 Ϫ 9xy Ϫ 3 y2 ϭ 5. Then, find , 18. __________________
the angle of rotation about the origin, to the nearest degree.
19. Solve the system algebraically. Round to the nearest tenth. 19. __________________
x2 ϩ y2 ϭ 9
2 x2 ϩ 3 y2 ϭ 18
20. Graph the solutions for the system of inequalities. 20.
(x Ϫ 1)2 ϩ 2( y ϩ 3)2 Ͻ 16
(x Ϫ 3)2 Յ Ϫ8( y ϩ 2)
Bonus Find the coordinates of the point(s) of Bonus: __________________
intersection of the graphs of Ϫᎏ23ᎏx ϩ y ϭ 0, Advanced Mathematical Concepts
x2 ϩ y2 ϭ 13, and y ϩ 1 ϭ ᎏ14ᎏ(x ϩ 2)2.
© Glencoe/McGraw-Hill 452
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Open-Ended Assessment
Instructions: Demonstrate your knowledge by giving a
clear, concise solution to each problem. Be sure to include all
relevant drawings and justify your answers. You may show
your solution in more than one way or investigate beyond the
requirements of the problem.
1. Consider the equation of a conic section written in the form
A x2 ϩ B y2 ϩ Cx ϩ D y ϩ E ϭ 0.
a. Explain how you can tell if the equation is that of a circle.
Write an equation of a circle whose center is not the origin.
Graph the equation.
b. Explain how you can tell if the equation is that of an ellipse.
Write an equation of an ellipse whose center is not the origin.
Graph the equation.
c. Explain how you can tell if the equation is that of a parabola.
Write an equation of a parabola with its vertex at (Ϫ1, 2).
Graph the equation.
d. Explain how you can tell if the equation is that of a hyperbola.
Write an equation of a hyperbola with a vertical transverse
axis.
e. Identify the graph of 3x2 Ϫ xy ϩ 2 y2 Ϫ 3 ϭ 0. Then find the
angle of rotation to the nearest degree.
2. a. Describe the graph of x2 Ϫ 4 y2 ϭ 0.
b. Graph the relation to verify your conjecture.
c. What conic section does this graph represent?
3. Give a real-world example of a conic section. Discuss how you
know the object is a conic section and analyze the conic section
if possible.
© Glencoe/McGraw-Hill 453 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 Mid-Chapter Test (Lessons 10-1 through 10-4)
Find the distance between each pair of points with the given coordinates. Then,
find the midpoint of the segment that has endpoints at the given coordinates.
1. __________________
1. (1, Ϫ4), (2, Ϫ9) 2. (s, Ϫt), (6 ϩ s, Ϫ5 Ϫ t) 2. __________________
3. Determine whether the quadrilateral ABCD with vertices 3. __________________
A(Ϫ2, 2), B(1, 3), C(4, Ϫ1), and D(1, Ϫ2) is a parallelogram. 4. __________________
4. Write the standard form of x2 Ϫ 6x ϩ y2 ϩ 4 y Ϫ 12 ϭ 0.
Then, graph the equation labeling the center.
5. Write the standard form of the equation of the circle that 5. __________________
passes through the points (Ϫ2, 16), (Ϫ2, 0), and (Ϫ32, 0).
Then, identify the center and radius. 6. __________________
7.
For the equation of each ellipse, find the coordinates of
the center, foci, and vertices. Then, graph the equation.
6. Write the standard form of 9x2 ϩ y2 ϩ 18x Ϫ 6y ϩ 9 ϭ 0.
Then, find the coordinates of the center, the foci, and
the vertices of the ellipse.
7. Graph the ellipse with equation ᎏ(x ϩ2ᎏ55)2 ϩ ᎏ( y Ϫ1ᎏ64)2 ϭ 1.
Label the center and vertices.
8. Write the equation of the 8. __________________
hyperbola graphed at
the right.
9. Find the coordinates of the center, the foci, and the vertices 9. __________________
for the hyperbola whose equation is 4y2 Ϫ 9x2 Ϫ 48y Ϫ
18x ϩ 99 ϭ 0. Then, find the equations of the asymptotes
and graph the equation.
© Glencoe/McGraw-Hill 454 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10, Quiz A (Lessons 10-1 and 10-2)
Find the distance between each pair of points with the given 1. __________________
coordinates. Then, find the midpoint of the segment that 2. __________________
has endpoints at the given coordinates.
1. (Ϫ2, 4), (5, Ϫ3) 2. (a, b), (a Ϫ 4, b ϩ 3)
3. Determine whether the quadrilateral ABCD with vertices 3. __________________
A(Ϫ1, 1), B(3, 3), C(3, 0), D(Ϫ1, Ϫ1) is a parallelogram. 4. __________________
4. Write the standard form of x2 Ϫ 6x ϩ y2 Ϫ 10y Ϫ 2 ϭ 0.
Then graph the equation, labeling the center.
5. Write the standard form of the equation of the circle that 5. __________________
passes through the points (2, 10), (2, 0), and (Ϫ10, 0).
Then, identify the center and radius.
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10, Quiz B (Lessons 10-3 and 10-4)
1. Write the standard form of 16x2 ϩ 4y2 Ϫ 96x ϩ 8y ϩ 84 ϭ 0. 1. __________________
Then, find the coordinates of the center, the foci, and the
vertices of the ellipse.
2. Graph the ellipse with equation ᎏ(x Ϫ6ᎏ46)2 ϩ ᎏ(y1ϩ0ᎏ01)2 ϭ 1. 2.
Label the center and vertices.
3. Write the equation of the 3. __________________
hyperbola shown in the
graph at the right.
4. Find the coordinates of the center, the foci, and the vertices 4.
for the hyperbola whose equation is 25x2 Ϫ 4y2 Ϫ 150x Ϫ
16y ϭ Ϫ109. Then, find the equations of the asymptotes
and graph the equation.
© Glencoe/McGraw-Hill 455 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10, Quiz C (Lessons 10-5 and 10-6)
1. Write the standard form of x2 Ϫ 4 x ϩ 8 y ϩ 12 ϭ 0. Identify 1. __________________
the coordinates of the focus and vertex, and the equations
of the directrix and axis of symmetry. Then, graph
the equation.
2. Write the equation of the parabola that has a focus at 2. __________________
(Ϫ2, 3) and whose directrix is given by the equation x ϭ 4.
3. Identify the conic section represented by the equation 3. __________________
4 x2 ϩ 25 y2 ϩ 16x ϩ 50 y Ϫ 59 ϭ 0. Then, write the
equation in standard form.
4. Find the rectangular equation of the curve whose parametric 4.
equations are x ϭ Ϫ3t 2 and y ϭ 2t, Ϫ2 Յ t Յ 2. Then graph
the equation, using arrows to indicate the orientation.
5. Find parametric equations for the equation x2 ϩ y2 ϭ 100. 5. __________________
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10, Quiz D (Lessons 10-7 and 10-8)
Identify the graph of each equation. Write an equation of the 1. __________________
translated or rotated graph in general form. 2. __________________
1. 5x2 Ϫ 8 y2 ϭ 40 for T(Ϫ3, 5) 2. 4 x2 ϩ 18 y2 ϭ 36, ϭ 135°
3. Identify the graph of x2 Ϫ 3 xy ϩ 2y2 ϩ 2 x Ϫ y ϩ 6 ϭ 0. 3. __________________
4. Solve the system x2 ϭ 25 Ϫ y2 and xy ϭ Ϫ12 algebraically. 4. __________________
5. Graph the solutions for the system of inequalities. 5.
x2 ϩ 9 y2 Ն 36
x2 Ϫ 2y Ͻ 4
© Glencoe/McGraw-Hill 456 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 SAT and ACT Practice
After working each problem, record the 6. In circle O, OA ϭ 6 and
correct answer on the answer sheet OෆෆA ⊥ ෆOෆB. Find the area
provided or use your own paper. of the shaded region.
A 2 units2
Multiple Choice B ( Ϫ 2) units2
1. Three points on a line are X, Y, and Z, C (6 Ϫ 9͙ෆ3) units2
in that order. If XZ Ϫ YZ ϭ 6, what is
the ratio ᎏXYᎏZZ ? D (9 Ϫ 18) units2
A 1 to 2
B 1 to 3 E (36 Ϫ 9͙ෆ3) units2
C 1 to 4
D 1 to 5 7. If 1 dozen pencils cost 1 dollar, how
E It cannot be determined from the many dollars will n pencils cost?
information given. A 12n
B ᎏ1nᎏ2
2. A train traveling 90 miles per hour for C ᎏ1n2ᎏ
D ᎏ121ᎏn
1 hour covers the same distance as a E It cannot be determined from the
information given.
train traveling 60 miles per hour for
8. On a map drawn to scale, 0.125 inch
how many hours? represents 10 miles. What is the
actual distance between two cities that
A ᎏ12ᎏ B ᎏ13ᎏ are 2.5 inches apart on the map?
D ᎏ32ᎏ A 12.5 mi
C ᎏ23ᎏ B 25 mi
E3 C 200 mi
D 250 mi
3. If x and y are integers and xy ϭ 48, E 1250 mi
then which of the following CANNOT
be true? 9. In the diagram below, if ᐉ͉͉m, then
A x ϩ y Ͻ 14 B x Ϫ y Ͼ 14 I. Є3 and Є4 are supplementary.
C x ϩ y ϭ 14 D ͉x ϩ y͉ Ͻ 14 II. mЄ2 ϭ mЄ10.
E ͉x ϩ y͉ Ͼ 14
III. mЄ6 ϭ mЄ8 ϩ mЄ9.
4. If x 0, then ᎏ(ϪϪ44ᎏxx3)3 ϭ
A Ϫ16 A I only
B II only
B Ϫ1 C III only
D II and III only
C3 E I and III only
D1
E 16
5. A framed picture is 5 feet by 8 feet,
including the frame. If the frame is
8 inches wide, what is the ratio of the
area of the frame to the area of the
framed picture, including the frame?
A 7 to 18
B 7 to 11
C 11 to 18
D 143 to 180
E 37 to 180
© Glencoe/McGraw-Hill 457 Advanced Mathematical Concepts
Chapter NAME _____________________________ DATE _______________ PERIOD ________
10 Chapter 10 SAT and ACT Practice (continued)
10. ෆPෆC and ෆABෆ intersect at point Q. 15. If Nancy earns d dollars in h hours,
mЄPQB ϭ (2z ϩ 80)؇, mЄBQC ϭ
(4x ϩ 3y)؇, mЄCQA ϭ 5w؇, and how many dollars will she earn in
mЄ AQP ϭ 2z؇. Find the value of w.
A 20 h ϩ 25 hours?
B 26
C 75 A ᎏ2h5ᎏd B d ϩ ᎏ25hᎏd
D 130
E It cannot be determined from the C 26d D ᎏh ϩdᎏh25
information given.
E None of these
11. Which point lies the greatest distance
from the origin? 16. If 6n cans f ill ᎏn2ᎏ cartons, how many
A (0, Ϫ9) cans does it take to fill 2 cartons?
B (Ϫ2, 9)
C (Ϫ7, Ϫ6) A 12 B 24n
D (8, 5)
E (Ϫ5, 7) C 24 D 6n2
12. The vertices of rectangle ABCD are the E ᎏ32ᎏ n2
points A(0, 0), B(8, 0), C(8, k), and
D(0, 5). What is the value of k? 17–18. Quantitative Comparison
A2 A if the quantity in Column A is
B3 greater
C4 B if the quantity in Column B is
D5 greater
E6 C if the two quantities are equal
D if the relationship cannot be
13. The measures of three exterior angles determined from the
of a quadrilateral are 37؇, 58؇, and 92؇. information given
What is the measure of the exterior
angle at the fourth vertex? Column A Column B
A 7؇
B 173؇ 17. Ratio of ᎏ13ᎏ to ᎏ51ᎏ Ratio of ᎏ52ᎏ to ᎏ13ᎏ
C 83؇
D 49؇ 18. Ratio of boys to Ratio of boys to
E 81؇ girls in a class
girls in a class with twice as many
14. The diagonals of parallelogram A BCD with half as many girls as boys
intersect at point O. If AO ϭ 2x ϩ 1
and AC ϭ 5x Ϫ 5, then AO ϭ boys as girls
A5
B7 19. Grid-In If 4 pounds of fertilizer cover
C 15 1500 square feet of lawn, how many
D 30 pounds of fertilizer are needed to cover
E It cannot be determined from the 2400 square feet?
information given.
20. Grid-In The distance between two
cities is 750 miles. How many inches
apart will the cities be on a map with a
scale of 1 inch ϭ 250 miles?
© Glencoe/McGraw-Hill 458 Advanced Mathematical Concepts