Chapter 10 Resource Masters

Chapter NAME _____________________________ DATE _______________ PERIOD ________

10 Chapter 10 Cumulative Review (Chapters 1-10)

1. Write a linear function that has no zero. 1. __________________
2. __________________
3 Ϫ2 5 Ϫ1 0 3. __________________
61 2 7 Ϫ4
΄ ΅ ΄ ΅2. Find BC if B ϭ 4. __________________
and C ϭ .

3. Consider the system of inequalities 3x ϩ 2y Ն 10,
x ϩ 3y Ն 9, x Ն 0, and y Ն 0. In a problem asking
you to find the minimum value of ƒ(x, y) ϭ x ϩ 3y,
state whether the situation is infeasible, has

alternate optimal solutions, or is unbounded.

4. Given ƒ(x) ϭ ᎏx Ϫ2ᎏ5 , find ƒϪ1(x). Then, state whether
ƒϪ1(x) is a function.

5. If y varies directly as the square of x, inversely as w, 5. __________________
inversely as the square of z, and y ϭ 2 when x ϭ 1,
w ϭ 4, and z ϭ Ϫ2, find y when x ϭ 3, w ϭ Ϫ6, and 6. __________________
z ϭ Ϫ3. 7. __________________

6. Solve ᎏxx Ϫϩᎏ13 ϭ ᎏ3xᎏ ϩ ᎏ12ᎏ.

x2 Ϫ 3x

7. Suppose ␪ is an angle in standard position whose
Ϫᎏ17ᎏ3 ,
terminal side lies in Quadrant II. If cos ␪ ϭ
find the value of csc ␪.

8. Solve ᭝ ABC if B ϭ 47Њ, C ϭ 68Њ, and b ϭ 29.2. 8. __________________

9. Find the linear velocity of the tip of an airplane propeller 9. __________________
that is 3 meters long and rotating 500 times per minute.
Give the velocity to the nearest meter per second.

10. Solve tan ␪ ϭ cot ␪ for 0Њ Յ ␪ < 360Њ. 10. __________________

11. Jason is riding his sled down a hill. If the hill is inclined 11. __________________
at an angle of 20Њ with the horizontal, find the force that 12. __________________
propels Jason down the hill if he weighs 151 pounds.

12. Find (͙ෆ3 ϩ i)5. Express the result in rectangular form.

13. Write the equation in standard form 13. __________________
of the ellipse graphed at the right.

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Practice Enrichment

Introduction to Analytic Geometry Mathematics and History: Hypatia

Find the distance between each pair of points with the given Hypatia (A.D. 370 –415) is the earliest woman mathematician whose
coordinates. Then find the midpoint of the segment that has life is well documented. Born in Alexandria, Egypt, she was widely
endpoints at the given coordinates. known for her keen intellect and extraordinary mathematical ability.
Students from Europe, Asia, and Africa flocked to the university at
1. (Ϫ2, 1), (3, 4) 2. (1, 1), (9, 7) Alexandria to attend her lectures on mathematics, astronomy,
philosophy, and mechanics.
͙3ෆ4ෆ; (0.5, 2.5) 10 ; (5, 4)
Hypatia wrote several major treatises in mathematics. Perhaps the
3. (3, Ϫ4),(5, 2) 4. (Ϫ1, 2), (5, 4) most significant of these was her commentary on the Arithmetica of Answers (Lesson 10-1)
Diophantus, a mathematician who lived and worked in Alexandria
2͙1ෆෆ0; (4, Ϫ1) 2͙ෆ1ෆ0; (2, 3) in the third century. In her commentary, Hypatia offered several
observations 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.

5. (Ϫ7, Ϫ4), (2, 8) 6. (Ϫ4, 10), (4, Ϫ5) In modern mathematics, the solutions of a Diophantine equation
are restricted to integers. In the exercises, you will explore some
15; (Ϫ2.5, 2) 17; (0, 2.5) questions involving simple Diophantine equations.

A3 Advanced Mathematical Concepts For each equation, find three solutions that consist of an ordered
pair of integers.

Determine whether the quadrilateral having vertices with the 1. 2x Ϫ y ϭ 7 2. x ϩ 3y ϭ 5
given coordinates is a parallelogram.
(1, Ϫ5), (0, Ϫ7), (Ϫ1, Ϫ9) (2, 1), (5, 0), (8, Ϫ1)

7. (4, 4), (2, Ϫ2), (Ϫ5, Ϫ1), (Ϫ3, 5) 8. (3, 5), (Ϫ1, 1), (Ϫ6, 2), (Ϫ3, 7) 3. 6x Ϫ 5y ϭ Ϫ8 4. Ϫ11x Ϫ 4y ϭ 6

yes no (2, 4), (Ϫ3, Ϫ2), (Ϫ8, Ϫ8) (2, Ϫ7), (Ϫ2, 4), (Ϫ6, 15)

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.

9. (4, Ϫ1), (2, Ϫ5), (Ϫ3, Ϫ3), (Ϫ1, 1) 10. (2, 6), (1, 2), (Ϫ4, 4), (Ϫ3, 9) Other integer pairs are of the form (x1 ϩ n и B, y1 Ϫ n и A),
where n is any nonzero integer.
yes no

11. Hiking Jenna and Maria are hiking to a campsite located at 6. Explain why the equation 3x ϩ 6y ϭ 7 has no solutions that are

(2, 1) on a map grid, where each side of a square represents integer pairs.
2.5 miles. If they start their hike at (Ϫ3, 1), how far must they
hike to reach the campsite? Rewrite 3x ϩ 6y ϭ 7 as 3(x ϩ 2y) ϭ 7. If x and y are
integers 7 would have to be an integral multiple of 3.
12.5 mi
7. True or false: Any line on the coordinate plane must pass through

at least one point whose coordinates are integers. Explain.

False; An equation like 3x ϩ 6y ϭ 7 has no integer-pair
solutions, so the graph of such an equation is a line that
passes through no point whose coordinates are integers.

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Practice Enrichment

Circles Spheres

Write the standard form of the equation of each circle described. The set of all points in three-dimensional space
Then graph the equation. that are a fixed distance r (the radius), from a
fixed point C (the center), is called a sphere. The
1. center at (3, 3) tangent to the x-axis 2. center at (2, Ϫ1), radius 4 equation below is an algebraic representation of
the sphere shown at the right.
( x Ϫ 3)2 ϩ ( y Ϫ 3)2 ϭ 9 ( x Ϫ 2)2 ϩ ( y ϩ 1)2 ϭ 16
(x – h) 2 ϩ ( y – k) 2 ϩ (z – l ) 2 ϭ r2

A line segment containing the center of a sphere Answers (Lesson 10-2)
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.

Write the standard form of each equation. Then graph the 1. If x 2 ϩ y2 – 4y ϩ z2 ϩ 2 z – 4 ϭ 0 is an equation of a
equation. sphere and (1, 4, –3) is one pole of the sphere, find the
coordinates of the opposite pole.
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
(–1, 0, 1)
( x Ϫ 4)2 ϩ ( y Ϫ 3)2 ϭ 4 ( xϩ 2)2 ϩ ( y Ϫ 1)2 ϭ ᎏ24ᎏ5
2. a. On the coordinate system at the right, sketch the
A4 Advanced Mathematical Concepts Write the standard form of the equation of the circle that passes sphere described by the equation x 2 ϩ y 2 ϩ z2 ϭ 9.
through the points with the given coordinates. Then identify the
center and radius. b. Is P(2, –2, –2) inside, outside, or on the sphere?

5. (Ϫ3, Ϫ2), (Ϫ2, Ϫ3), (Ϫ4, Ϫ3) 6. (0, Ϫ1), (2, Ϫ3), (4, Ϫ1) outside

(x ϩ 3)2 ϩ ( y ϩ 3)2 ϭ 1; (x Ϫ 2)2 ϩ ( y ϩ 1)2 ϭ 4; c. Describe a way to tell if a point with coordinates
(Ϫ3, Ϫ3); 1 (2, Ϫ1); 2 P(a, b, c) is inside, outside, or on the sphere with
equation x 2 ϩ y 2 ϩ z2 ϭ r2.
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 a2 ϩ b 2 ϩ c 2 Ͻ r 2 : inside the sphere
is the equation of the circle that circumscribes the square? a2 ϩ b 2 ϩ c 2 ϭ r 2 : on the sphere
a2 ϩ b 2 ϩ c 2 Ͼ r 2 : outside the sphere
x2 ϩ y2 ϭ 8
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.

8␲ units; 64␲ square units;
ᎏ256␲ cubic units

3

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.

a cylinder

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Practice Enrichment

Ellipses Superellipses

Write the equation of each ellipse in standard form. Then find the The circle and the ellipse are members of an interesting family of
coordinates of its foci. curves that were first studied by the French physicist and
1. 2. mathematician Gabriel Lame′ (1795-1870). The general equation for
the family is

nϩΈ Έ Έ Έxy n ϭ 1, with a 0, b 0, and n Ͼ 0.
ᎏa ᎏb

For even values of n greater than 2, the curves are called
superellipses.

ᎏ( y 2Ϫᎏ53)2 ϩ ᎏ(x Ϫ9ᎏ2)2 ϭ 1; ᎏ(x Ϫ3ᎏ64)2 ϩ ᎏ( y 1Ϫᎏ62)2 ϭ 1; 1. Consider two curves that are not Answers (Lesson 10-3)
(2, Ϫ1), (2, 7) superellipses. Graph each equation on the
(4 Ϫ 2͙ෆ5, 2), (4 ϩ 2͙ෆ5, 2) grid at the right. State the type of curve
produced each time.

Έ Έ Έ Έa. ᎏ2x 2 ϩ ᎏ2y 2 ϭ 1 circle

For the equation of each ellipse, find the coordinates of the center, Έ Έ Έ Έb. ᎏ3x 2 ϩ ᎏ2y 2 ϭ 1 ellipse
foci, and vertices. Then graph the equation.

A5 Advanced Mathematical Concepts 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 2. In each of the following cases you are given

center: (1, 2); center: (1, 5); values of a, b, and n to use in the general
foci: (1, 9),
foci: (1Ϯ͙ෆ5, 2) (1, 1) equation. Write the resulting equation.

vertices: vertices: Then graph. Sketch each graph on the grid
(Ϫ2, 2), (1, 4), (1, 10), (1, 0),
(4, 2), (1, 0) (4, 5), (Ϫ2, 5) at the right.

ᎏx 4 ᎏy 4
2 3
ϩ ϭ1
Έ Έ Έ Έa. a ϭ 2, b ϭ 3, n ϭ 4

x 6 y 6
ᎏ2 ᎏ3
ϩ ϭ
Έ Έ Έ Έb. a ϭ 2, b ϭ 3, n ϭ 6 1

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 ᎏx 8 ᎏy 8
parallel to the y-axis, and one vertex is (Ϫ2, Ϫ7), and a ϭ 5. Έ Έ Έ Έc. a ϭ 2, b ϭ 3, n ϭ 82 ϩ 3 1
at (1, 8), and b ϭ 3. ϭ
ᎏ( y 2ϩ5ᎏ3)2 ϩ ᎏ(x ϩ9ᎏ2)2 ϭ 1
ᎏ( y 2Ϫ5ᎏ3)2 ϩ ᎏ(x Ϫ9ᎏ1)2 ϭ 1 Έ Έ Έ Έ3. What shape will the graph of ᎏ2x n ϩ ᎏ3y n ϭ 1

7. Construction A semi elliptical arch is used to design a approximate for greater and greater even,
headboard for a bed frame. The headboard will have a height whole-number values of n?
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? a rectangle that is 6 units long
and 4 units wide, centered at
1.5 ft from the center the origin

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Practice Enrichment

Hyperbolas Moving Foci

For each equation, find the coordinates of the center, foci, and vertices, and Recall that the equation of a hyperbola with center
the equations of the asymptotes of its graph. Then graph the equation. at the origin and horizontal transverse axis has the

1. x2 Ϫ 4y2 Ϫ 4 x ϩ 24y Ϫ 36 ϭ 0 2. y2 Ϫ 4 x2 ϩ 8 x ϭ 20 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 ± b x. Such a hyperbola is shown at the right.
ᎏa

What happens to the shape of the graph as c grows
very large or very small?

A6 Advanced Mathematical Concepts center: (2, 3); foci (2Ϯ͙ෆ5, 3); center: (1, 0); foci: (1, Ϯ2 ͙ෆ5); Refer to the hyperbola described above. Answers (Lesson 10-4)

vertices: (0, 3), (4, 3); vertices: (1, Ϯ4) 1. Write a convincing argument to show that as c approaches 0, the
asymptotes: y Ϫ 3 ϭ Ϯ ᎏ12ᎏ( x Ϫ 2) asymptotes: y ϭ Ϯ2 (x Ϫ 1)
foci, the vertices, and the center of the hyperbola become the
Write the equation of each hyperbola. 4.
3. same point.

ᎏ(x Ϫ4ᎏ1)2 Ϫ ᎏ( y ϩ9ᎏ2)2 ϭ 1 ᎏ(x Ϫ1ᎏ1)2 Ϫ ᎏ( y Ϫ4ᎏ3)2 ϭ 1 Since 0 Ͻ a Ͻ c, as c approaches 0, a
approaches 0. So the x-coordinates of the foci
5. Write an equation of the hyperbola for which the length of the and vertices approach 0, which is the
x-coordinate of the center. Since the
transverse axis is 8 units, and the foci are at (6, 0) and (Ϫ4, 0). y-coordinates are equal, the points become the
same.
ᎏ( x Ϫ1ᎏ61)2 Ϫ y2 ϭ 1
ᎏ9ᎏ 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
6. Environmental Noise Two neighbors who live one mile apart to smaller and smaller values of c.) Describe the changes in the

hear an explosion while they are talking on the telephone. One graphs. What shape do the graphs approach as c approaches 0?

neighbor hears the explosion two seconds before the other. If The asymptotes remain the same, but the
branches become sharper near the vertices.
sound travels at 1100 feet per second, determine the equation of The graphs approach the lines y ϭ x and
y ϭ –x.
the hyperbola on which the explosion was located.
3. Suppose a is held fixed and c approaches infinity. How does the
ᎏ1,21x0ᎏ2,000 Ϫ y2 ϭ 1
ᎏ5,759ᎏ,600 graph of the hyperbola change?

The vertices remain at (Ϯa, 0), but the branches
become more vertical. The graphs approach
the vertical lines x ϭ –a and x ϭ a.

4. Suppose b is held fixed and c approaches infinity. How does the

graph of the hyperbola change?

The vertices recede to infinity and the branches become
flatter and farther from the center. As c approaches infinity,
the graphs tend to disappear.

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Practice Enrichment

Parabolas Tilted Parabolas

For the equation of each parabola, find the coordinates of the The diagram at the right shows a fixed point F(1, 1)
vertex and focus, and the equations of the directrix and axis of and a line d whose equation is y ϭ –x Ϫ 2. If P(x, y)
symmetry. Then graph the equation. satisfies the condition that PD ϭ PF, then P is on a
parabola. Our objective is to find an equation for the
1. x2 Ϫ 2 x Ϫ 8 y ϩ 17 ϭ 0 2. y2 ϩ 6 y ϩ 9 ϭ 12 Ϫ 12x 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.
vertex: (1, 2); focus: (1, 4); vertex: (1, Ϫ3); focus: (Ϫ2, Ϫ3);
directrix: y ϭ 0; directrix: x ϭ 4; To do this, first find an equation for the line m through
axis of symmetry: x ϭ 1 axis of symmetry: y ϭ Ϫ3 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 Answers (Lesson 10-5)
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.

x Ϫy ϩ (b Ϫ a) ϭ 0

Write the equation of the parabola that meets each set of 2. Use the equations for lines m and d to show that the coordinates
conditions. Then graph the equation.
ᎏx Ϫ yᎏϪ 2 ᎏy Ϫ xᎏϪ 2
2 2
A7 Advanced Mathematical Concepts ΂ ΃of point D are D(a, b) ϭ D , .

3. The vertex is at (Ϫ2, 4) and 4. The focus is at (2, 1), and the From the equation for line m,
the focus is at (Ϫ2, 3). equation of the directrix is x ϭ Ϫ2.
–a ϩ b ϭ –x ϩ y. From the equation for d,
( x ϩ 2)2 ϭ Ϫ4( y Ϫ 4) ( y Ϫ 1)2 ϭ 8x
a ϩ b ϭ –2. Subtract to get a ϭ ᎏx Ϫ 2yᎏϪ 2 .
Add to get b ϭ ᎏy Ϫ xᎏϪ 2 .
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.

x 2 Ϫ 2xy ϩ y 2 Ϫ 8x Ϫ 8y ϭ 0

5. Satellite Dish Suppose the receiver in a parabolic dish antenna 4. a. Every parabola has an axis of symmetry. Find an equation for
is 2 feet from the vertex and is located at the focus. Assume that the axis of symmetry of the parabola described above. Justify
the vertex is at the origin and that the dish is pointed upward. your answer.
Find an equation that models a cross section
of the dish. y ϭ x, since y ϭ x contains F (1,1) and is
perpendicular to d.
x2 ϭ 8y
b. Use your answer from part a to find the coordinates of the
vertex of the parabola. Justify your answer.

(0,0), since (0,0) is midway between point F
and line d.

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Practice Enrichment

Rectangular and Parametric Forms of Conic Sections Polar Graphs of Conics

Identify the conic section represented by each equation. Then A conic is the locus of all points such that the ratio e of the y
write the equation in standard form and graph the equation. Ox
distance from a fixed point F and a fixed line d is constant.
e > 1: hyperbola
1. x2 Ϫ 4y ϩ 4 ϭ 0 2. x2 ϩ y2 Ϫ 6x Ϫ 6 y Ϫ 18 ϭ 0 FP ϭ e
ᎏDP y
parabola; x2 ϭ 4( y Ϫ 1) circle;
( x Ϫ 3)2 ϩ ( y Ϫ 3)2 ϭ 36 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
Answers (Lesson 10-6)
yy

3. 4 x2 Ϫ y2 Ϫ 8x ϩ 6 y ϭ 9 4. 9x2 ϩ 5y2 ϩ 18x ϭ 36 Ox Ox

hyperbola; ᎏ(x Ϫ1ᎏ1)2 Ϫ ᎏ( y Ϫ4ᎏ3)2 ϭ 1 ellipse; ᎏ( x ϩ5ᎏ1)2 ϩ y2 ϭ 1 Ox
ᎏ9ᎏ

A8 Advanced Mathematical Concepts Graph each relation and identify the conic.

1. r ϭ ᎏ1 – 4cᎏos ␪ parabola 2. r ϭ ᎏ2 – 4cᎏos ␪ ellipse

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␲ 3. r ϭ ᎏ2 ϩ4sᎏin ␪ ellipse 4. r ϭ ᎏ1 ϩ 24ᎏsin ␪ hyperbola

x2 ϩ y2 ϭ 9 ᎏ1x6ᎏ2 ϩ y2 ϭ 1
ᎏ2ᎏ5

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Practice Enrichment

Transformations of Conics Graphing with Addition of y-Coordinates

Identify the graph of each equation. Write an equation of the Equations of parabolas, ellipses, and hyperbolas
translated or rotated graph in general form. that are “tipped” with respect to the x- and y-axes
are more difficult to graph than the equations you
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) have been studying.

ellipse parabola Often, however, you can use the graphs of two
simpler equations to graph a more complicated
2 x2 ϩ 5y2 ϩ 8x Ϫ 10y ϩ 4 ϭ 0 2x2 Ϫ 8x Ϫ y ϩ 8 ϭ 0 equation. For example, the graph of the ellipse in
the diagram at the right is obtained by adding the
3. xy ϭ 1, ␪ ϭ ᎏ␲4ᎏ y-coordinate of each point on the circle and the
y-coordinate of the corresponding point of the line.
hyperbola
4. x2 Ϫ 4y ϭ 0, ␪ ϭ 90° Answers (Lesson 10-7)
Ϫᎏ21ᎏ x2 ϩ ᎏ21ᎏ y 2 Ϫ 1 ϭ 0
parabola
y2 ϩ 4x ϭ 0

A9 Advanced Mathematical Concepts 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 Graph each equation. State the type of curve for each graph.

circle hyperbola 1. y ϭ 6 Ϫ x Ϯ ͙ෆ4ෆϪෆxෆ2 ellipse 2. y ϭ x Ϯ ͙ෆx parabola
45° Ϫ41°

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

parabola ellipse

Ϫ37° Ϫ27°

9. Communications Suppose the orientation of a satellite Use a separate sheet of graph paper to graph these equations. State the type of
dish that monitors radio waves is modeled by the equation curve for each graph.
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? 3. y ϭ 2x Ϯ ͙7ෆෆϩෆ6xෆෆϪෆxෆ2 ellipse; 4. y ϭ Ϫ2x Ϯ ͙–ෆ2ෆෆx parabola;
See students' graphs. See students' graphs.
45°

© Glencoe/McGraw-Hill 436 Advanced Mathematical Concepts © Glencoe/McGraw-Hill 437 Advanced Mathematical Concepts

© Glencoe/McGraw-Hill 10-8 NAME _____________________________ DATE _______________ PERIOD ________ 10-8 NAME _____________________________ DATE _______________ PERIOD ________

Practice Enrichment

Systems of Second-Degree Equations and Inequalities Intersections of Circles

Solve each system of equations algebraically. Round to the Many interesting problems involving circles can be
solved by using a system of equations. Consider the
nearest tenth. Check the solutions by graphing each system. following problem.

1. 2x Ϫ y ϭ 8 2. x2 Ϫ y2 ϭ 4 Find an equation for the straight line that contains
x2 ϩ y2 ϭ 9 the two points of intersection of two intersecting
yϭ1 circles whose equations are given.
no real solutions
(Ϫ2.2, 1) and (2.2, 1) 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.

3. xy ϭ 4 4. x2 ϩ y2 ϭ 4 1. Circle A has equation x 2 ϩ y2 ϭ 1 and circle B has Answers (Lesson 10-8)
equation (x – 3)2 ϩ y2 ϭ 1. Use a sketch to show that the
x2 ϭ y2 ϩ 1 4 x2 ϩ 9 y2 ϭ 36 circles do not intersect. Use an algebraic argument to
show that circles A and B do not intersect.
(2.1, 1.9) and (Ϫ2.1, Ϫ1.9) (0, Ϫ2) and (0, 2)
The distance between the centers is 3.
A10 Since the sum of the radii is 2, there is 1
unit of space between the circles. Thus,
Graph each system of inequalities. 6. (x Ϫ 1)2 ϩ ( y Ϫ 2)2 Ͻ 9 the circles do not intersect.
4( y ϩ 1)2 ϩ x2 Յ 16
5. 3 Ն ( y Ϫ 1)2 ϩ 2x 2. Circle A has equation (x – 2)2 ϩ ( y ϩ 1)2 ϭ 16 and
y Ն Ϫ3x ϩ 1 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.

10x Ϫ 2y ϩ 11 ϭ 0

Advanced Mathematical Concepts 7. Sales Vincent’s Pizzeria reduced prices for large specialty 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
pizzas by $5 for 1 week in March. In the previous week, sales for tangent. Justify your answer.

large specialty pizzas totaled $400. During the sale week, the The distance between the centers is ͙5ෆ Ϸ 2.24.
Since the sum of the radii is 2 ϩ 2͙ෆ2 Ϸ 4.83,
number of large pizzas sold increased by 20 and total sales the circles overlap and meet in two points; they

amounted to $600. Write a system of second-degree equations to are not tangent.

model this situation. Find the regular price and the sale price

of large specialty pizzas. xy ϭ 400, ( x ϩ 20) ( y Ϫ 5) ϭ 600;
regular price: $20.00, sale price: $15.00

© Glencoe/McGraw-Hill 439 Advanced Mathematical Concepts © Glencoe/McGraw-Hill 440 Advanced Mathematical Concepts

Chapter 10 Answer Key

Form 1A Form 1B

Page 441 Page 442 Page 443 Page 444

1. B 13. A 1. B 13. B

2. A 2. C

3. C 3. A

4. B 14. C 4. C 14. D
5. C 5. B 15. B
6. C 15. D 6. A 16. D
16. B

7. B 17. C 7. B 17. A
8. A 18. C 8. A 18. A

9. C 9. B

10. C

11. A 19. D 10. B
12. B
20. B 19. D
Bonus: C 11. A
12. D 20. B
Bonus: B

© Glencoe/McGraw-Hill A11 Advanced Mathematical Concepts

Chapter 10 Answer Key

Form 1C Form 2A

Page 445 Page 446 Page 447 Page 448

1. D 13. C 1. ͙ෆ5ෆ8 ( y Ϫ 1)2 ϭ x ϩ 6;
2. no 10. x ϭ Ϫᎏ24ᎏ5 ; y ϭ 1

2. B

3. B (x ϩ 2)2 ϩ 11. (x ϩ 2)2 ϭ Ϫᎏ38ᎏ( y Ϫ 1)
4. A 3. ( y Ϫ 3)2 ϭ 41

5. C (x Ϫ 4)2 ϩ
6. C 4. ( y ϩ 1)2 ϭ 25
7. B
8. B 14. D 5. ΂x Ϫ ΃ᎏ25ᎏ 2 ϩ ΂y Ϫ ΃ᎏ29ᎏ 2 ϭ 27 12. ᎏ( y ϩ25ᎏ3)2 Ϫ ᎏ(x Ϫ5ᎏ64)2 ϭ 1
9. D 15. D
16. A 13. hyperbola
10. B 17. A
11. A 14.(ᎏx Ϫᎏ41ᎏ 2)2 ϩ ᎏ( y ϩ4ᎏ3)2 ϭ 1
18. B
12. C 19. C Sample answer:
20. B
x ϭ 4 cos t Ϫ 1,

6.ᎏ(x ϩ8ᎏ4)2 ϩ ᎏ( y Ϫ1ᎏ21)2 ϭ 1 y ϭ ᎏ͙2ᎏෆ2 sin t ϩ 3,
0° Յ t Յ 360°
15.

7. ᎏ(x Ϫ9ᎏ4)2 ϩ ᎏ( y ϩ25ᎏ3)2 ϭ 1 parabola;

16. x2 ϩ 4x ϩ 12y ϩ 70 ϭ 0

ellipse; ΂5 Ϫ ͙ෆ3΃x2 Ϫ

17. ΂2 Ϫ 2͙ෆ3΃xy ϩ ΂7 ϩ ͙ෆ3΃y2 Ϫ 8 ϭ 0

18. hyperbola, 19°

8.ᎏ( y ϩ3ᎏ3)2 Ϫ ᎏ(x ϩ9ᎏ2)2 ϭ 1 19. (Ϯ2.7, 0.5)
20.
( y Ϫ 1)2 Ϫ ᎏ(x ϩ1ᎏ26)2 ϭ 1
ᎏ8ᎏ
asymptotes:

9. y Ϫ 1 ϭ Ϯ ᎏ͙3ᎏෆ6 (x ϩ 6)

Bonus: A Bonus: (1, 3)

© Glencoe/McGraw-Hill A12 Advanced Mathematical Concepts

Chapter 10 Answer Key

Form 2B Form 2C

Page 449 Page 450 Page 451 Page 452

1. 4͙5ෆ ( y Ϫ 1)2 ϭ 2( x Ϫ 2); 1. 5 ( y ϩ 2)2 ϭ 4(x Ϫ 2);

10. x ϭ ᎏ32ᎏ; y ϭ 1 10. x ϭ 1; y ϭ Ϫ2

2. yes 2. no

(x Ϫ 1)2 ϩ (x ϩ 4)2 ϩ
3. ( y ϩ 3)2 ϭ 16 3. ( y Ϫ 3)2 ϭ 25

(x ϩ 3)2 ϩ 11. ( y ϩ 5)2 ϭ Ϫ40(x Ϫ 2) (x Ϫ 5)2 ϭ
4. ( y Ϫ 2)2 ϭ 10 11. Ϫ4( y ϩ 1)
4. x2 ϩ y2 ϭ 8
(x ϩ 3)2 ϩ
5. ( y Ϫ 7)2 ϭ 100 12. ᎏx1ᎏ2 ϩ ᎏy9ᎏ2 ϭ 1 (x Ϫ 4)2 ϩ 12. ᎏ͙2ᎏෆ2
5. ( y ϩ 2)2 ϭ 4

13. hyperbola

13. hyperbola 14. ᎏx4ᎏ2 ϩ y2 ϭ 1
14. ᎏx1ᎏ2 ϩ ᎏy4ᎏ2 ϭ 1 ᎏ9ᎏ

Sample answer: Sample answer:
x ϭ 4 cos t, y ϭ 6 sin t,
x ϭ 2͙2ෆ cos t,
15. 0Њ Յ t Յ 360Њ y ϭ 2͙2ෆ sin t,

15. 0° Յ t Յ 360Њ

6. ᎏ(x Ϫ9ᎏ3)2 ϩ ᎏ( y ϩ4ᎏ1)2 ϭ 1 6. ᎏ(y Ϫ9ᎏ2)2 ϩ (x ϩ 3)2 ϭ 1
ᎏ4ᎏ
0
parabola; parabola;

16. x2 ϩ 2x Ϫ 2y Ϫ 3 ϭ 0 16. y2 ϩ 8x ϩ 16y ϩ 72 ϭ 0

7. ᎏ(x Ϫ4ᎏ2)2 ϩ ᎏ( y ϩ2ᎏ03)2 ϭ 1 ellipse; 7. ᎏ(x ϩ9ᎏ1)2 ϩ ᎏ( y Ϫ1ᎏ3)2 ϭ 1 ellipse; 9(xЈ)2 ϩ 2xЈyЈ
11(xЈ)2 Ϫ 6͙3ෆ xЈyЈ ϩ
8. ᎏ( y Ϫ9ᎏ4)2 Ϫ ᎏ(x 2ϩᎏ52)2 ϭ 1 17. ϩ 9 (yЈ)2 Ϫ 40 ϭ 0
17. 17 (yЈ)2 Ϫ 80 ϭ 0
18. hyperbola, Ϫ27Њ ᎏ(x Ϫ1ᎏ62)2 Ϫ ᎏ( y ϩ4ᎏ1)2 ϭ 1 18. hyperbola, Ϫ26Њ
asymptotes:
19. no solution 19. (3, 0), (Ϫ3, 0)
9. y ϩ 1 ϭ Ϯ ᎏ12ᎏ(x Ϫ 2)
8. ᎏ(x Ϫ1ᎏ65)2 Ϫ ᎏ( y ϩ9ᎏ4)2 ϭ 1 20. 20.

(x ϩ 3)2 Ϫ ( y Ϫ 2)2 ϭ 1
ᎏ1ᎏ ᎏ4ᎏ
asymptotes:

9. y Ϫ 2 ϭ Ϯ2(x ϩ 3)

© Glencoe/McGraw-Hill Bonus: (1, Ϫ2) Bonus: (2, 3)

A13 Advanced Mathematical Concepts

Chapter 10 Answer Key

CHAPTER 10 SCORING RUBRIC

Level Specific Criteria

3 Superior • Shows thorough understanding of the concepts circle, ellipse,
parabola, hyperbola, center, vertex, and angle of rotation.

• Uses appropriate strategies to identify equations of
conic sections.

• Computations are correct.
• Written explanations are exemplary.
• Real-world example of conic section is appropriate and

makes sense.
• Graphs are accurate and appropriate.
• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts circle, ellipse,
with Minor parabola, hyperbola, center, vertex, and angle of rotation.
Flaws
• Uses appropriate strategies to identify equations of conic
sections.

• Computations are mostly correct.
• Written explanations are effective.
• Real-world example of conic section is appropriate and

makes sense.
• Graphs are mostly accurate and appropriate.
• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts circle, ellipse,
Satisfactory, parabola, hyperbola, center, vertex, and angle of rotation.
with Serious
• May not use appropriate strategies to identify equations
Flaws of conic sections.

• Computations are mostly correct.
• Written explanations are satisfactory.
• Real-world example of conic section is mostly appropriate

and sensible.
• Graphs are mostly accurate and appropriate.
• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts circle,
ellipse, parabola, hyperbola, center, vertex, and angle of
rotation.

• May not use appropriate strategies to identify equations
of conic sections.

• Computations are incorrect.
• Written explanations are not satisfactory.
• Real-world example of conic section is not appropriate or

sensible.
• Graphs are not accurate or appropriate.
• Does not satisfy requirements of problems.

© Glencoe/McGraw-Hill A14 Advanced Mathematical Concepts

Chapter 10 Answer Key

Open-Ended Assessment

Page 453 1e. The graph is an ellipse since

1a. The equation is a circle if A ϭ B. (Ϫ1)2 Ϫ 4(3)(2) Ͻ 0.
Sample answer: tan 2␪ ϭ ᎏ3ϪϪᎏ12 ϭ Ϫ1, 2␪ ϭ Ϫ45Њ,
(x Ϫ 1)2 ϩ ( y Ϫ 2)2 ϭ 4 ␪ ϭ Ϫᎏ␲8ᎏ, or Ϫ22.5Њ.

1b. The equation is an ellipse if A B 2a. The graph of x2 Ϫ 4y2 ϭ 0 is two
and A and B have the same sign. lines of slope ᎏ12ᎏ and slope Ϫᎏ12ᎏ that
intersect at the origin.
Sample answer: x2 Ϫ 4y2 ϭ 0
ᎏ(x ϩ4ᎏ2)2 ϩ ᎏ( y Ϫ1ᎏ1)2 ϭ 1 x2 ϭ 4y2
|x| ϭ 2|y|
ᎏ|2xᎏ| ϭ |y|
ᎏ2xᎏ ϭ y or ᎏϪ2ᎏx ϭ y

2b.

1c. The equation is a parabola when A 2c. degenerate hyperbola
or B is zero, but not both.
Sample answer: y Ϫ 2 ϭ 4(x ϩ 1)2 3. Sample answer: Most lamps with
circular shades shine a cone of
light. When this light cone strikes a
nearby wall, the resulting shape is a
hyperbola. The hyperbola is formed
by the cone of light intersecting the
plane of the wall.

1d. The equation is a hyperbola if A and

B have opposite signs. Sample
answer: ᎏy4ᎏ2 Ϫ ᎏx1ᎏ2 ϭ 1

© Glencoe/McGraw-Hill A15 Advanced Mathematical Concepts

Chapter 10 Answer Key

Mid-Chapter Test Quiz A Quiz C
Page 454 Page 455 Page 456

1. ͙2ෆෆ6, ΂ᎏ32ᎏ, Ϫᎏ12ᎏ3 ΃ ΂ ΃1. 7͙ෆ2, ᎏ32ᎏ, ᎏ12ᎏ 1. (x Ϫ 2)2 ϭ Ϫ8( y ϩ 1), (2, Ϫ3);
2. ͙6ෆෆ1, ΂3 ϩ s, ᎏϪ5 2Ϫᎏ2t ΃
2. ΂ ΃5; a Ϫ 2, ᎏ2b2ϩᎏ3 (2, Ϫ1); y ϭ 1; x ϭ 2

3. yes 3. no

4. (x Ϫ 3)2 ϩ ( y ϩ 2)2 ϭ 25 4. (x Ϫ 3)2 ϩ ( y Ϫ 5)2 ϭ 36 2. ( y Ϫ 3)2 ϭ Ϫ12(x Ϫ 1)

(x ϩ 4)2 ϩ ( y Ϫ 5)2 ϭ 61; 3. ellipse; ᎏ(x ϩ2ᎏ52)2 ϩ ᎏ( y ϩ4ᎏ1)2 ϭ 1
4. y2 ϭ Ϫᎏ43ᎏ x
5. (Ϫ4, 5); ͙ෆ61ෆ Ϸ 7.8

(x ϩ 17)2 ϩ ( y Ϫ 8)2 ϭ
5. 289; (Ϫ17, 8); 17

ᎏ( yϪ9ᎏ3)2 ϩ ᎏ( x ϩ1ᎏ1)2 ϭ 1; Quiz B
(Ϫ1, 3); (Ϫ1, 3 Ϯ 2͙2ෆ);
Page 455
6. (Ϫ1, 6), (Ϫ1, 0), (0, 3), (Ϫ2, 3)
( y ϩ 1)2 ϩ (x Ϫ 3)2 ϭ 1;
ᎏ1ᎏ6 ᎏ4ᎏ
(3, Ϫ1); (3, Ϫ1 Ϯ 2͙3ෆ); Sample answer:
1. (3, 3), (3, Ϫ5), (5, Ϫ1), (1, Ϫ1) x ϭ 10 cos t, y ϭ 10 sin t
5. 0 Յ t Յ 2␲
7.
Quiz D
8. ᎏ(x Ϫ16ᎏ3)2 Ϫ ᎏ( y Ϫ9ᎏ2)2 ϭ 1 2. Page 456

(Ϫ1, 6); (Ϫ1, 6 Ϯ ͙ෆ1ෆ3); 3. ᎏ( y Ϫ4ᎏ3)2 Ϫ ᎏ(x Ϫ9ᎏ1)2 ϭ 1 hyperbola;
(Ϫ1, 9), (Ϫ1, 3); (3, Ϫ2); ΂3 Ϯ͙ෆ29ෆ, Ϫ2΃;
9. y Ϫ 6 ϭ Ϯ ᎏ23ᎏ (x ϩ 1) 1. 5x2 Ϫ 8y2 ϩ 30x ϩ 80y Ϫ 195 ϭ 0
(5, Ϫ2), (1, Ϫ2); ellipse; 11(xЈ)2 ϩ

4. y ϩ 2 ϭ Ϯᎏ25ᎏ(x Ϫ 3) 2. 14xЈyЈ ϩ 11( yЈ)2 Ϫ 36 ϭ 0

3. hyperbola

4. (3, Ϫ4), (Ϫ3, 4)

5.

© Glencoe/McGraw-Hill A16 Advanced Mathematical Concepts

Chapter 10 Answer Key

SAT/ACT Practice Cumulative Review

Page 457 Page 458 Page 459
1. Sample answer: ƒ(x) ϭ 3
1. E 10. B
11 Ϫ17 8
2. D 11. D 32 1 Ϫ4
΄ ΅2.

3. Alternate optimal solutions

3. D 12. D 4. ƒϪ1(x) ϭ ᎏ2xᎏ ϩ 5; yes
4. E 13. B 5. Ϫᎏ136ᎏ
5. A 14. C
6. D 15. B 6. Ϫ1
7. B 16. C
8. C 17. A 7. ᎏ13͙6ᎏ0ෆ3ෆ0
9. E 18. C
19. 6.4 8. A ϭ 65°, a ϭ 36.2, c ϭ 37.0

9. 79 m/s

10. 45°, 135°
11. 51.6 lb

12. Ϫ16͙3ෆ ϩ 16i

13. ᎏ(x Ϫ9ᎏ1)2 ϩ ( y Ϫ 2)2 ϭ 1
ᎏ4ᎏ

© Glencoe/McGraw-Hill 20. 3 Advanced Mathematical Concepts

A17

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