Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10 Anticipation Guide 10-1 Lesson Reading Guide
Conic Sections Midpoint and Distance Formulas
Step 1 Before you begin Chapter 10 Get Ready for the Lesson
• Read each statement. Chapter Resources Read the introduction to Lesson 10-1 in your textbook. Answers (Anticipation Guide and Lesson 10-1)
How do you find distances on a road map?
• Decide whether you Agree (A) or Disagree (D) with the statement.
Sample answer: Use the scale of miles on the map. You might also use a
• Write A or D in the first column OR if you are not sure whether you agree or disagree, ruler.
write NS (Not Sure).
Read the Lesson
STEP 1 Statement STEP 2 Lesson 10-1
A, D, or NS A or D 1. a. Write the coordinates of the midpoint of a segment with endpoints (x1, y1) and (x2, y2).
A1 Glencoe Algebra 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. To find the midpoint between two points on a coordinate A Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ᎏx1 ϩ2 x2 , ᎏy1 ϩ2 y2
plane, find the means of the coordinates of the points.
D b. Explain how to find the midpoint of a segment if you know the coordinates of the
2. All points on a parabola are the same distance from a given A endpoints. Do not use subscripts in your explanation.
point and the x-axis. D
D Sample answer: To find the x-coordinate of the midpoint, add the
3. In the equation for a parabola, y ϭ a (x Ϫ h )2 ϩ k, if a Ͻ 0 the A x-coordinates of the endpoints and divide by two. To find the
parabola opens downward. A y-coordinate of the midpoint, do the same with the y-coordinates of
D the endpoints.
4. If the equation of a circle is (x ϩ 3)2 ϩ (y ϩ 6)2 ϭ 16, then the A
center of the circle is at the point (3, 6). 2. a. Write an expression for the distance between two points with coordinates (x1, y1) and
D
5. A tangent line to a circle is a line that intersects the circle in (x2, y2). ͙(ෆx2 Ϫ xෆ1)2 ϩෆ(y2 Ϫෆy1)2
two points.
b. Explain how to find the distance between two points. Do not use subscripts in your
6. The foci of an ellipse always lie on the major axis. explanation.
7. The asymptotes of a graph of a hyperbola are lines that the Sample answer: Find the difference between the x-coordinates and
graph approaches but never reaches. square it. Find the difference between the y-coordinates and square it.
Add the squares. Then find the square root of the sum.
8. Conic sections are formed by the intersection of two cones.
3. Consider the segment connecting the points (Ϫ3, 5) and (9, 11).
9. When the equation of a conic section is written in standard
form it is possible to name which conic section the equation a. Find the midpoint of this segment. (3, 8)
represents. b. Find the length of the segment. Write your answer in simplified radical form. 6͙5ෆ
10. No solution can be found for a system of equations containing Remember What You Learned
both linear and quadratic equations.
4. How can the “mid” in midpoint help you remember the midpoint formula?
Step 2 After you complete Chapter 10
Sample answer: The midpoint is the point in the middle of a segment.
• Reread each statement and complete the last column by entering an A or a D. It is halfway between the endpoints. The coordinates of the midpoint
are found by finding the average of the two x-coordinates (add them
• Did any of your opinions about the statements change from the first column? and divide by 2) and the average of the two y-coordinates.
• For those statements that you mark with a D, use a piece of paper to write an example
of why you disagree.
Chapter 10 3 Glencoe Algebra 2 Chapter 10 5 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-1 Study Guide and Intervention 10-1 Study Guide and Intervention (continued)
Midpoint and Distance Formulas Midpoint and Distance Formulas
The Midpoint Formula The Distance Formula
Midpoint Formula The midpoint M of a segment with endpoints (x1, y1) and (x2, y2) is ᎏx1 ϩ2 x2 , ᎏy1 ϩ2 y2 . Distance Formula The distance between two points (x1, y1) and (x2, y2) is given by
d ϭ ͙ෆ(x2 Ϫ xෆ1)2 ϩෆ(y2 Ϫෆy1)2.
Example 1 Find the midpoint of the Example 2 A diameter AෆBෆ of a circle Example 1 What is the distance between (8, Ϫ2) and (Ϫ6, Ϫ8)?
line segment with endpoints at has endpoints A(5, Ϫ11) and B(Ϫ7, 6).
(4, Ϫ7) and (Ϫ2, 3). What are the coordinates of the center d ϭ ͙(ෆx2 Ϫ xෆ1)2 ϩ (ෆy2 Ϫ ෆy1)2 Distance Formula
of the circle? ϭ ͙ෆ(Ϫ6 Ϫෆ8)2 ϩෆ[Ϫ8 Ϫෆ(Ϫ2)]2ෆ Let (x1, y1) ϭ (8, Ϫ2) and (x2, y2) ϭ (Ϫ6, Ϫ8).
ᎏx1 ϩ2 x2 , ᎏy1 ϩ2 y2 ϭ ᎏ4 ϩ2(ᎏϪ2) , ᎏϪ72ϩ 3 ϭ ͙(ෆϪ14)ෆ2 ϩ (Ϫ6ෆ)2 Subtract.
The center of the circle is the midpoint of all Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
ϭ ᎏ22 , ᎏϪ24 or (1, Ϫ2) of its diameters. ϭ ͙1ෆ96 ϩෆ36 or ͙2ෆ32 Simplify.
Lesson 10-1
The midpoint of the segment is (1, Ϫ2). ᎏx1 ϩ2 x2 , ᎏy1 ϩ2 y2 ϭ ᎏ5 ϩ2(ᎏϪ7) , ᎏϪ112ᎏϩ 6 The distance between the points is ͙ෆ232 or about 15.2 units. Answers (Lesson 10-1)
ϭ ᎏϪ22 , ᎏϪ25 or Ϫ1, Ϫ2ᎏ12 Example 2 Find the perimeter and area of square PQRS with vertices P(Ϫ4, 1),
Q(Ϫ2, 7), R(4, 5), and S(2, Ϫ1).
The circle has center Ϫ1, Ϫ2ᎏ12 .
Find the length of one side to find the perimeter and the area. Choose PෆෆQ.
A2 Glencoe Algebra 2 Exercises d ϭ ͙ෆ(x2 Ϫ ෆx1)2 ϩ (ෆy2 Ϫ ෆy1)2 Distance Formula
Find the midpoint of each line segment with endpoints at the given coordinates. ϭ ͙[ෆϪ4 Ϫෆ(Ϫ2)]2ෆϩ (1ෆϪ 7)2 Let (x1, y1) ϭ (Ϫ4, 1) and (x2, y2) ϭ (Ϫ2, 7).
ϭ ͙ෆ(Ϫ2)2 ෆϩ (Ϫ6ෆ)2 Subtract.
1. (12, 7) and (Ϫ2, 11) 2. (Ϫ8, Ϫ3) and (10, 9) 3. (4, 15) and (10, 1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ϭ ͙ෆ40 or 2͙1ෆ0 Simplify.
(5, 9) (1, 3) (7, 8)
Since one side of the square is 2͙ෆ10, the perimeter is 8͙ෆ10 units. The area is (2͙1ෆ0)2, or
4. (Ϫ3, Ϫ3) and (3, 3) 5. (15, 6) and (12, 14) 6. (22, Ϫ8) and (Ϫ10, 6)
40 units2.
(0, 0) (13.5, 10) (6, Ϫ1)
Exercises
7. (3, 5) and (Ϫ6, 11) 8. (8, Ϫ15) and (Ϫ7, 13) 9. (2.5, Ϫ6.1) and (7.9, 13.7) Find the distance between each pair of points with the given coordinates.
Ϫᎏ32 , 8 ᎏ21 , Ϫ1 (5.2, 3.8) 1. (3, 7) and (Ϫ1, 4) 2. (Ϫ2, Ϫ10) and (10, Ϫ5) 3. (6, Ϫ6) and (Ϫ2, 0)
10. (Ϫ7, Ϫ6) and (Ϫ1, 24) 11. (3, Ϫ10) and (30, Ϫ20) 12. (Ϫ9, 1.7) and (Ϫ11, 1.3) 5 units 13 units 10 units
(Ϫ4, 9) ᎏ323 , Ϫ15 (Ϫ10, 1.5) 4. (7, 2) and (4, Ϫ1) 5. (Ϫ5, Ϫ2) and (3, 4) 6. (11, 5) and (16, 9)
13. Segment ෆMNෆ has midpoint P. If M has coordinates (14, Ϫ3) and P has coordinates 3͙ෆ2 units 10 units ͙4ෆ1 units
(Ϫ8, 6), what are the coordinates of N? (Ϫ30, 15) 7. (Ϫ3, 4) and (6, Ϫ11) 8. (13, 9) and (11, 15) 9. (Ϫ15, Ϫ7) and (2, 12)
3͙3ෆ4 units 2͙ෆ10 units 5͙ෆ26 units
14. Circle R has a diameter ෆSTෆ. If R has coordinates (Ϫ4, Ϫ8) and S has coordinates (1, 4), 10. Rectangle ABCD has vertices A(1, 4), B(3, 1), C(Ϫ3, Ϫ2), and D(Ϫ5, 1). Find the
what are the coordinates of T? (Ϫ9, Ϫ20) perimeter and area of ABCD. 2͙1ෆ3 ϩ 6͙5ෆ units; 3͙6ෆ5 units2
15. Segment ෆADෆ has midpoint B, and BෆෆD has midpoint C. If A has coordinates (Ϫ5, 4) and 11. Circle R has diameter ෆSෆT with endpoints S(4, 5) and T(Ϫ2, Ϫ3). What are the
C has coordinates (10, 11), what are the coordinates of B and D? circumference and area of the circle? (Express your answer in terms of .)
B is 5, 8ᎏ23 , D is 15, 13ᎏ31 . 10 units; 25 units2
Chapter 10 6 Glencoe Algebra 2 Chapter 10 7 Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-1 Skills Practice 10-1 Practice
Midpoint and Distance Formulas Midpoint and Distance Formulas
Find the midpoint of each line segment with endpoints at the given coordinates. Find the midpoint of each line segment with endpoints at the given coordinates.
1. (4, Ϫ1), (Ϫ4, 1) (0, 0) 2. (Ϫ1, 4), (5, 2) (2, 3) 1. (8, Ϫ3), (Ϫ6, Ϫ11) (1, Ϫ7) 2. (Ϫ14, 5), (10, 6) Ϫ2, ᎏ121
3. (3, 4), (5, 4) (4, 4) 4. (6, 2), (2, Ϫ1) 4, ᎏ12 3. (Ϫ7, Ϫ6), (1, Ϫ2) (Ϫ3, Ϫ4) 4. (8, Ϫ2), (8, Ϫ8) (8, Ϫ5)
6. (Ϫ3, 5), (Ϫ3, Ϫ8) Ϫ3, Ϫᎏ32
5. (3, 9), (Ϫ2, Ϫ3) ᎏ12 , 3 5. (9, Ϫ4), (1, Ϫ1) 5, Ϫᎏ52 6. (3, 3), (4, 9) ᎏ72 , 6
7. (4, Ϫ2), (3, Ϫ7) ᎏ72 , Ϫᎏ92 8. (6, 7), (4, 4) 5, ᎏ121
10. (5, Ϫ2), (3, 7) 4, ᎏ52
9. (Ϫ4, Ϫ2), (Ϫ8, 2) (Ϫ6, 0) 12. (Ϫ9, Ϫ8), (8, 3) Ϫᎏ12 , Ϫᎏ25
14. Ϫᎏ13 , 6 , ᎏ23 , 4 ᎏ16 , 5
11. (Ϫ6, 3), (Ϫ5, Ϫ7) Ϫᎏ121 , Ϫ2 16. ᎏ18 , ᎏ12 , Ϫᎏ58 , Ϫᎏ21 Ϫᎏ14 , 0
13. (2.6, Ϫ4.7), (8.4, 2.5) (5.5, Ϫ1.1)
15. (Ϫ2.5, Ϫ4.2), (8.1, 4.2) (2.8, 0)
7. (3, 2), (Ϫ5, 0) (Ϫ1, 1) 8. (3, Ϫ4), (5, 2) (4, Ϫ1) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9. (Ϫ5, Ϫ9), (5, 4) 0, Ϫᎏ25 10. (Ϫ11, 14), (0, 4) Ϫᎏ121 , 9 Lesson 10-1 Answers (Lesson 10-1)
11. (3, Ϫ6), (Ϫ8, Ϫ3) Ϫᎏ52 , Ϫᎏ29 12. (0, 10), (Ϫ2, Ϫ5) Ϫ1, ᎏ25
A3 Glencoe Algebra 2 Find the distance between each pair of points with the given coordinates. Find the distance between each pair of points with the given coordinates.
13. (4, 12), (Ϫ1, 0) 13 units 14. (7, 7), (Ϫ5, Ϫ2) 15 units Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 17. (5, 2), (2, Ϫ2) 5 units 18. (Ϫ2, Ϫ4), (4, 4) 10 units
19. (Ϫ3, 8), (Ϫ1, Ϫ5) ͙ෆ173 units 20. (0, 1), (9, Ϫ6) ͙1ෆ30 units
15. (Ϫ1, 4), (1, 4) 2 units 16. (11, 11), (8, 15) 5 units
21. (Ϫ5, 6), (Ϫ6, 6) 1 unit 22. (Ϫ3, 5), (12, Ϫ3) 17 units
17. (1, Ϫ6), (7, 2) 10 units 18. (3, Ϫ5), (3, 4) 9 units 23. (Ϫ2, Ϫ3), (9, 3) ͙1ෆ57 units 24. (Ϫ9, Ϫ8), (Ϫ7, 8) 2͙ෆ65 units
19. (2, 3), (3, 5) ͙ෆ5 units 20. (Ϫ4, 3), (Ϫ1, 7) 5 units 25. (9, 3), (9, Ϫ2) 5 units 26. (Ϫ1, Ϫ7), (0, 6) ͙1ෆ70 units
27. (10, Ϫ3), (Ϫ2, Ϫ8) 13 units 28. (Ϫ0.5, Ϫ6), (1.5, 0) 2͙ෆ10 units
21. (Ϫ5, Ϫ5), (3, 10) 17 units 22. (3, 9), (Ϫ2, Ϫ3) 13 units 29. ᎏ25 , ᎏ35 , 1, ᎏ57 1 unit 30. (Ϫ4͙2ෆ, Ϫ͙5ෆ), (Ϫ5͙2ෆ, 4͙5ෆ) ͙ෆ127 units
23. (6, Ϫ2), (Ϫ1, 3) ͙7ෆ4 units 24. (Ϫ4, 1), (2, Ϫ4) ͙6ෆ1 units 31. GEOMETRY Circle O has a diameter AෆBෆ. If A is at (Ϫ6, Ϫ2) and B is at (Ϫ3, 4), find the
25. (0, Ϫ3), (4, 1) 4͙ෆ2 units 26. (Ϫ5, Ϫ6), (2, 0) ͙8ෆ5 units center of the circle and the length of its diameter. Ϫᎏ29 , 1 ; 3͙5ෆ units
32. GEOMETRY Find the perimeter of a triangle with vertices at (1, Ϫ3), (Ϫ4, 9), and (Ϫ2, 1).
18 ϩ 2͙ෆ17 units
Chapter 10 8 Glencoe Algebra 2 Chapter 10 9 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-1 Word Problem Practice 10-1 Enrichment
Midpoint and Distance Formulas
1. EXHIBITS Museum planners want to 4. AIRPLANES A grid is superimposed Distance Between Points in Space
place a statue directly in the center of on a map of Texas. Dallas has
their Special Exhibits Room. Suppose coordinates (200, 5) and Amarillo has The Distance Formula and Midpoint Formula on the coordinate z
the room is placed on a coordinate plane coordinates (Ϫ100, 208). If each unit plane is derived from the Pythagorean Theorem a2 ϩ b2 ϭ c2. (0, 1, 3)
as shown. What are the coordinates of represents 1 mile, how long will it take
the center of this room? a plane flying at an average speed of In three dimensions, the coordinate grid contains the x-axis and Oy
410 miles per hour to fly directly from the y-axis, as in two-dimensional geometry, and also a z-axis. (1, 3, 0)
y Dallas to Amarillo? Round your answer An example of a line segment drawn on a three-dimensional
to the nearest tenth of an hour. coordinate grid is shown at the right.
0.9 hour
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The three-dimensional distance formula is much like the one for
TRAVEL For Exercises 5 and 6, use the two dimensions. The distance from A (x1, y1, z1) to B (x2, y2, z2) can
following information and the figure Lesson 10-1
below. be found using d ϭ ͙ෆ(x1 Ϫ xෆ2) 2 ϩෆ(y1 Ϫෆy2) 2 ϩෆ(z1 Ϫෆz2) 2.
x The Martinez family is planning a trip x
from their home in Fort Lauderdale to
O Tallahassee. They plan to stop overnight
at a location about halfway between the
(2, 4) two cities. Example Find the distance between the points (1, 3, 0) and (0, 1, 3). Answers (Lesson 10-1)
2. WALKING Laura starts at the origin. Tallahassee y Jacksonville d ϭ ͙(ෆ1 Ϫ 0ෆ)2 ϩ (3ෆϪ 1ෆ)2 ϩ (ෆ0 Ϫ 3ෆ)2 Replace (x1, y1, z1) with (1, 3, 0) and (x2, y2, z2) with (0, 1, 3).
She walks 8 units to the right and then
12 units up. How far away from the Gainsville 5 Daytona Beach d ϭ ͙ෆ1 ϩ 4ෆϩ 9 or ͙ෆ14 Simplify.
origin is she? Round your answer to the
A4 Glencoe Algebra 2 nearest tenth. Exercises
14.4 units
1. Find the distance between each pair of points A (1, 3, Ϫ2) and B (4, 2, 1).
3. SURVEILLANCE A grid is superimposed Land O’Lakes Orlando Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
on a map of the area directly d ͙ ؍1ෆ9
surrounding the home of a suspect. St. Petersburg Plant City
Detectives want to position themselves 2. Find the distance between each pair of points C (5, 3, 2) and D (0, 1, 7).
on opposite sides of the suspect’s Ϫ5 Tampa O 5 x
house. Coordinates are assigned to the d ͙ ؍5ෆ4
suspect’s home. Unit A is positioned at Fort Lauderdale
(Ϫ1, 6) on the coordinate plane. Where 0 100 mi Ϫ5 Miami 3. Find the distance between each pair of points E (Ϫ2, Ϫ1, 6) and F (Ϫ1, 3, 2).
should Unit B be located so that the
suspect’s home is centered between the 5. What are the coordinates of the point d ͙ ؍ෆ33
two units? halfway between Tallahassee and Fort
Lauderdale? Which of the cities on the 4. Use what you know about the midpoint formula for a segment graphed on a regular
y map is closest to this point? coordinate grid to make a conjecture about the formula for finding the coordinates of a
Unit A midpoint in three-dimensions.
(0, 1); Land O’Lakes
Suspect’s home Midpoint ؍ᎏx1 ؉2 x2 , ᎏy1 ؉2 y2 , ᎏz1 ؉2 z2 .
6. How many miles is it from Fort
O x Lauderdale to Tallahassee? Round your 5. Find the midpoint for each segment in Exercises 1–3.
answer to the nearest mile.
(5/2, 5/2, ؊1/2), (5/2, 2, 9/2), (؊3/2, 1, 4)
521 mi
(9, 0)
Chapter 10 10 Glencoe Algebra 2 Chapter 10 11 Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-2 Lesson Reading Guide 10-2 Study Guide and Intervention
Parabolas Parabolas
Get Ready for the Lesson Equations of Parabolas A parabola is a curve consisting of all points in the
Read the introduction to Lesson 10-2 in your textbook. coordinate plane that are the same distance from a given point (the focus) and a given line
Name at least two reflective objects that might have the shape of a parabola. (the directrix). The following chart summarizes important information about parabolas.
Sample answer: telescope mirror, satellite dish Standard Form of Equation y ϭ a(x Ϫ h)2 ϩ k x ϭ a(y Ϫ k)2 ϩ h
Axis of Symmetry xϭh yϭk
Read the Lesson Vertex (h, k ) (h, k )
Focus
1. In the parabola shown in the graph, the point (2, Ϫ2) is called y h, k ϩ ᎏ41a h ϩ ᎏ41a , k
xϭ2 Directrix
the vertex and the point (2, 0) is called the Direction of Opening y ϭ k Ϫ ᎏ41a x ϭ h Ϫ ᎏ41a
(2, 0) Length of Latus Rectum upward if a Ͼ 0, downward if a Ͻ 0 right if a Ͼ 0, left if a Ͻ 0
focus . The line y ϭ Ϫ4 is called the Ox
⏐ ⏐ᎏa1 units ⏐ ⏐ᎏa1 units Answers (Lesson 10-2)
(2, –2)
directrix , and the line x ϭ 2 is called the Example
y ϭ –4
axis of symmetry . Identify the coordinates of the vertex and focus, the equations of
the axis of symmetry and directrix, and the direction of opening of the parabola
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
with equation y ϭ 2x2 Ϫ 12x Ϫ 25.
Lesson 10-2
2. a. Write the standard form of the equation of a parabola that opens upward or y ϭ 2x2 Ϫ 12x Ϫ 25 Original equation
y ϭ 2(x2 Ϫ 6x) Ϫ 25 Factor 2 from the x-terms.
downward. y ϭ a (x Ϫ h)2 ϩ k y ϭ 2(x2 Ϫ 6x ϩ ■) Ϫ 25 Ϫ 2(■) Complete the square on the right side.
y ϭ 2(x2 Ϫ 6x ϩ 9) Ϫ 25 Ϫ 2(9) The 9 added to complete the square is multiplied by 2.
A5 Glencoe Algebra 2 b. The parabola opens downward if a Ͻ 0 and opens upward if a Ͼ 0 . The y ϭ 2(x Ϫ 3)2 Ϫ 43 Write in standard form.
equation of the axis of symmetry is x ϭ h , and the coordinates of the vertex are Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The vertex of this parabola is located at (3, Ϫ43), the focus is located at 3, Ϫ42ᎏ78 , the
(h, k) . Ϫ43ᎏ18 .
3. A parabola has equation x ϭ Ϫᎏ81 ( y Ϫ 2)2 ϩ 4. This parabola opens to the left . equation of the axis of symmetry is x ϭ 3, and the equation of the directrix is y ϭ
The parabola opens upward.
It has vertex (4, 2) and focus (2, 2) . The directrix is x ϭ 6 . The length Exercises
of the latus rectum is 8 units. Identify the coordinates of the vertex and focus, the equations of the axis of
symmetry and directrix, and the direction of opening of the parabola with the
Remember What You Learned given equation.
4. How can the way in which you plot points in a rectangular coordinate system help you to 1. y ϭ x2 ϩ 6x Ϫ 4 2. y ϭ 8x Ϫ 2x2 ϩ 10 3. x ϭ y2 Ϫ 8y ϩ 6
remember what the sign of a tells you about the direction in which a parabola opens?
(Ϫ3, Ϫ13), (2, 18), 2, 17ᎏ81 , (Ϫ10, 4), Ϫ9ᎏ43 , 4,
Sample answer: In plotting points, a positive x-coordinate tells you to
move to the right and a negative x-coordinate tells you to move to the Ϫ3, Ϫ12ᎏ34 , x ϭ Ϫ3, x ϭ 2, y ϭ 18ᎏ81 , y ϭ 4, x ϭ Ϫ10ᎏ14 ,
left. This is like a parabola whose equation is of the form “x ϭ …”; it
opens to the right if a Ͼ 0 and to the left if a Ͻ 0. Likewise, a positive y ϭ Ϫ13ᎏ41 , up down right
y-coordinate tells you to move up and a negative y-coordinate tells you
to move down. This is like a parabola whose equation is of the form Write an equation of each parabola described below.
“y ϭ …”; it opens upward if a Ͼ 0 and downward if a Ͻ 0.
4. focus (Ϫ2, 3), directrix x ϭ Ϫ2ᎏ112 5. vertex (5, 1), focus 4ᎏ1112 , 1
x ϭ 6(y Ϫ 3)2 Ϫ 2ᎏ214 x ϭ Ϫ3(y Ϫ 1)2 ϩ 5
Chapter 10 12 Glencoe Algebra 2 Chapter 10 13 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-2 Study Guide and Intervention (continued) 10-2 Skills Practice
Parabolas Parabolas
Graph Parabolas To graph an equation for a parabola, first put the given equation in Write each equation in standard form.
standard form. 1. y ϭ x2 ϩ 2x ϩ 2 2. y ϭ x2 Ϫ 2x ϩ 4 3. y ϭ x2 ϩ 4x ϩ 1
y ϭ a(x Ϫ h)2 ϩ k for a parabola opening up or down, or y ϭ [x Ϫ (Ϫ1)]2 ϩ 1 y ϭ (x Ϫ 1)2 ϩ 3 y ϭ [x Ϫ (Ϫ2)]2 ϩ (Ϫ3)
x ϭ a(y Ϫ k)2 ϩ h for a parabola opening to the left or right
Identify the coordinates of the vertex and focus, the equations of the axis of
Use the values of a, h, and k to determine the vertex, focus, axis of symmetry, and length of symmetry and directrix, and the direction of opening of the parabola with the
the latus rectum. The vertex and the endpoints of the latus rectum give three points on the given equation. Then find the length of the latus rectum and graph the parabola.
parabola. If you need more points to plot an accurate graph, substitute values for points
near the vertex.
Example Graph y ϭ ᎏ13 (x Ϫ 1)2 ϩ 2. 4. y ϭ (x Ϫ 2)2 5. x ϭ (y Ϫ 2)2 ϩ 3 6. y ϭ Ϫ(x ϩ 3)2 ϩ 4
y y y
In the equation, a ϭ ᎏ31 , h ϭ 1, k ϭ 2. Answers (Lesson 10-2)
The parabola opens up, since a Ͼ 0. y Ox Ox
vertex: (1, 2) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.vertex: (3, 2);vertex: (Ϫ3, 4);
axis of symmetry: x ϭ 1 Ox Lesson 10-2 focus: 3ᎏ14 , 2 ; focus: Ϫ3, 3ᎏ43 ;
1 1, 2ᎏ43 vertex: (2, 0); axis of symmetry: axis of symmetry:
1, focus:2ϩᎏ or y ϭ 2; x ϭ Ϫ3;
4 ᎏ31ᎏ focus: 2, ᎏ41 ; directrix: x ϭ 2ᎏ34 ; directrix: y ϭ 4ᎏ14 ;
opens right; opens down;
A6 Glencoe Algebra 2 ⏐ ⏐length of latus rectum: ᎏᎏ113ᎏ or 3 units Ox axis of symmetry: latus rectum: 1 unit latus rectum: 1 unit
x ϭ 2;
endpoints of latus rectum: 2ᎏ12 , 2ᎏ34 , Ϫᎏ21 , 2ᎏ43 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. directrix: y ϭ Ϫᎏ41 ;
opens up;
Exercises latus rectum: 1 unit
The coordinates of the focus and the equation of the directrix of a parabola are Write an equation for each parabola described below. Then draw the graph.
given. Write an equation for each parabola and draw its graph.
1. (3, 5), y ϭ 1 2. (4, Ϫ4), yϭ Ϫ6 3. (5, Ϫ1), x ϭ 3 7. vertex (0, 0), 8. vertex (5, 1), 9. vertex (1, 3),
directrix x ϭ ᎏ87
y y y focus 0, Ϫᎏ112 focus 5, ᎏ45
x ϭ 2(y Ϫ 3)2 ϩ 1
y ϭ Ϫ3x 2 y ϭ (x Ϫ 5)2 ϩ 1
y
y y
O x Ox
Ox
Ox y ϭ ᎏ14 (x Ϫ 4)2 Ϫ 5 x ϭ ᎏ41 (y ϩ 1)2 ϩ 4 Ox Ox
y ϭ ᎏ18 (x Ϫ 3)2 ϩ 3
Chapter 10 14 Glencoe Algebra 2 Chapter 10 15 Glencoe Algebra 2
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-2 Practice 10-2 Word Problem Practice
Parabolas Parabolas
Write each equation in standard form. 1. PROJECTILE A projectile follows the 4. TELESCOPES An astronomer is working
graph of the parabola y ϭ Ϫ ᎏ32 x2 ϩ 6x. with a large reflecting telescope. The
1. y ϭ 2x2 Ϫ 12x ϩ 19 2. y ϭ ᎏ12 x2 ϩ 3x ϩ ᎏ21 3. y ϭ Ϫ3x2 Ϫ 12x Ϫ 7 Sketch the path of the projectile by reflecting mirror in the telescope has
graphing the parabola. the parabolic cross section shown in
y ϭ 2(x Ϫ 3)2 ϩ 1 y ϭ ᎏ21 [x Ϫ (Ϫ3)]2 ϩ (Ϫ4) y ϭ Ϫ3[x Ϫ (Ϫ2)]2 ϩ 5 the graph whose equation is given by
y y ϭ ᎏ18 (x Ϫ 4)2 ϩ 2. Each unit represents
Identify the coordinates of the vertex and focus, the equations of the axis of 6 1 meter. The astronomer is standing at
symmetry and directrix, and the direction of opening of the parabola with the the origin. How far from the focus of
given equation. Then find the length of the latus rectum and graph the parabola. O 4x the parabola is the point on the mirror
directly over the astronomer’s head?
4. y ϭ (x Ϫ 4)2 ϩ 3 5. x ϭ Ϫᎏ31 y2 ϩ 1 6. x ϭ 3(y ϩ 1)2 Ϫ 3
y
y yy
6
Ox Ox 2. COMMUNICATION David has just Answers (Lesson 10-2)
made a large parabolic dish whose
Ox cross section is based on the graph of Cross section of
the parabola y ϭ 0.25x2. Each unit mirror surface
represents one foot and the diameter
of his dish is 4 feet. He wants to make a x
listening device by placing a microphoneCopyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.O4
at the focus of the parabola. Where
vertex: (4, 3); vertex: (1, 0); vertex: (Ϫ3, Ϫ1); should the microphone be placed? Lesson 10-24m
1 foot away from the vertex along
A7 Glencoe Algebra 2 focus: 4, 3ᎏ41 ; focus: ᎏ14 , 0 ; focus: Ϫ2ᎏ1121 , Ϫ1 ; Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. the axis (at the point (0, 1) with BRIDGES For Exercises 5 and 6, use the
respect to the graph) following information.
axis: x ϭ 4; axis: y ϭ 0; axis: y ϭ Ϫ1; Part of the Sydney Harbor Bridge in
directrix: y ϭ 2ᎏ43 ; directrix: x ϭ 1ᎏ43 ; directrix: x ϭ Ϫ3ᎏ112 ; 3. BRIDGES A bridge is in the shape Sydney, Australia, can be modeled by
opens up; opens left; opens right; of a parabola that opens downward. a parabolic arch. If each unit corresponds
latus rectum: 1 unit latus rectum: 3 units latus rectum: ᎏ13 unit The equation of the parabola to model to 10 meters, the arch would pass through
the arch of the bridge is given by the points at (Ϫ25, 5), (0, 10), and (25, 5).
Write an equation for each parabola described below. Then draw the graph. y ϭ Ϫ ᎏ2x42 ϩ ᎏ65 x ϩ ᎏ161 , where each unit
5. Write the equation of the parabola to
7. vertex (0, Ϫ4), 8. vertex (Ϫ2, 1), 9. vertex (1, 3), is equivalent to 1 yard. The x-axis is the model the arch.
directrix x ϭ Ϫ3 axis of symmetry x ϭ 1, ground level. What is the maximum
focus 0, Ϫ3ᎏ78 latus rectum: 2 units, height of the bridge above the ground? y ؍؊ ᎏ1215 x2 ؉ 10
x ϭ ᎏ14 (y Ϫ 1)2 Ϫ 2 aϽ0
y ϭ 2x2 Ϫ 4 6 yd 6. Identify the coordinates of the focus of
y y ϭ Ϫᎏ21 (x Ϫ 1)2 ϩ 3 this parabola.
y
y (0, ؊21.25)
Ox Ox
Ox
10. TELEVISION Write the equation in the form y ϭ ax2 for a satellite dish. Assume that the
bottom of the upward-facing dish passes through (0, 0) and that the distance from the
bottom to the focus point is 8 inches. y ϭ ᎏ312 x2
Chapter 10 16 Glencoe Algebra 2 Chapter 10 17 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-2 Enrichment 10-2 Spreadsheet Activity
Limits Parabolas form of equation y ϭ a(x Ϫ h)2 ϩ k x ϭ a(y Ϫ k)2 ϩ h
vertex (h, k) (h, k)
Sequences of numbers with a rational expression for the general term often You have learned many of the axis of symmetry xϭh yϭk
approach some number as a finite limit. For example, the reciprocals of the characteristics of parabolas focus
positive integers approach 0 as n gets larger and larger. This is written using with vertical and horizontal ( )h, k ϩ ᎏ41aᎏ ( )h ϩ ᎏ41aᎏ, k
the notation shown below. The symbol ∞ stands for infinity and n → ∞ means axes of symmetry. The directrix
that n is getting larger and larger, or “n goes to infinity.” information is summarized y ϭ k Ϫ ᎏ41aᎏ x ϭ h Ϫ ᎏ41aᎏ
in the table at the right. You direction of opening upward if a Ͼ 0, right if a Ͼ 0, left
1, ᎏ21ᎏ, ᎏ31ᎏ, ᎏ41ᎏ, … , ᎏn1ᎏ, … lim ᎏn1ᎏ ϭ 0 can use what you know to downward if a Ͻ 0
create a spreadsheet to length of latus if a Ͻ 0
n→∞ analyze given equations of rectum | |ᎏ1aᎏ units
parabolas. | |ᎏ1aᎏ units
Example Find lim ᎏ(n ϩnᎏ21)2 Exercises Answers (Lesson 10-2)
n→ϱ The spreadsheet below uses the equation of a parabola in the form
y ϭ a(x Ϫ h)2 ϩ k or x ϭ a(y Ϫ k)2 ϩ h to find information about the
It is not immediately apparent whether the sequence approaches a limit or parabola. x or y is entered in Column D and the values of a, h, and k
are entered into Columns A, B, and C respectively.
not. But notice what happens if we divide the numerator and denominator of Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
the general term by n2.
Lesson 10-2
ᎏ(n ϩnᎏ21)2 ϭ ᎏn2 ϩ n2ᎏ2n ϩ 1 ABCDE F G H I J
ϭ ᎏᎏnnᎏ22 ϩ ᎏᎏ2nnnnᎏᎏᎏ222 ϩ ᎏn1ᎏ2
A8 Glencoe Algebra 2 a h k y ϭ or Vertex Length of Axis of Focus Directrix Direction
x ϭ Latus Symmetry of Opening
Rectum
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1 (units)
2 0.25 3.00 Ϫ1 x 3, Ϫ1 4.00 y ϭ Ϫ1 4, Ϫ1 x ϭ 2 Right
ϭ ᎏ1 ϩ ᎏn2ᎏ1ᎏϩ ᎏn1ᎏ2 3 3.00 Ϫ4.00 2.00 y Ϫ4, 2 0.33 x ϭ Ϫ4 Ϫ4, 2.08 y ϭ 1.92 Upward
4 Ϫ1.00 3.00 2.00 y 3, 2 1.00 x ϭ 3 3, 1.75 y ϭ 2.25 Downward
5
Sheet 1 Sheet 2 Sheet 3
The two fractions in the denominator will approach a limit of 0 as n gets 1. Which row represents the equation y ϭ 3x2 ϩ 24x ϩ 50? row 3
very large, so the entire expression approaches a limit of 1.
2. Write the standard form of the equation represented by row 2.
Exercises
x ϭ ᎏ41ᎏ (y ϩ 1)2 ϩ 3
Find the following limits. 3. What formula should be used in cell F2? 1/ABS(A2)
1. lim ᎏnn34ϩϪᎏ56n 0 2. lim ᎏ1 nϪᎏ2 n 0 4. Find the vertex, length of latus rectum, axis of symmetry, focus, directrix,
and direction of opening of a parabola with equation (y Ϫ 8)2 ϭ Ϫ4(x Ϫ 4).
n→∞ n→∞
(8, 4); 4; y ϭ 4; (7, 4); x ϭ 9; left
3. lim ᎏ2(n2ϩn ϩᎏ1) 1ϩ 1 1 4. lim ᎏ21nϪϩᎏ3n1 Ϫᎏ32ᎏ
n→∞ n→∞
Chapter 10 18 Glencoe Algebra 2 Chapter 10 19 Glencoe Algebra 2
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-3 Lesson Reading Guide 10-3 Study Guide and Intervention
Circles Circles
Get Ready for the Lesson Equations of Circles The equation of a circle with center (h, k) and radius r units is
Read the introduction to Lesson 10-3 in your textbook. (x Ϫ h)2 ϩ ( y Ϫ k)2 ϭ r2.
A large home improvement chain is planning to enter a new metropolitan area and needs to Example Write an equation for a circle if the endpoints of a diameter are at
select locations for its stores. Market research has shown that potential customers are
willing to travel up to 12 miles to shop at one of their stores. How can circles help the (Ϫ4, 5) and (6, Ϫ3).
managers decide where to place their store?
Use the midpoint formula to find the center of the circle.
Sample answer: A store will draw customers who live inside a circle with
center at the store and a radius of 12 miles. The management should select (h, k) ϭ ᎏx1 ϩ2 x2 , ᎏy1 ϩ2 y2 Midpoint formula
locations for which as many people as possible live within a circle of radius
12 miles around one of the stores. ϭ ᎏϪ42ϩ 6 , ᎏ5 ϩ2(ᎏϪ3) (x1, y1) ϭ (Ϫ4, 5), (x2, y2) ϭ (6, Ϫ3)
Read the Lesson ϭ ᎏ22 , ᎏ22 or (1, 1) Simplify.
1. a. Write the equation of the circle with center (h, k) and radius r. Use the coordinates of the center and one endpoint of the diameter to find the radius. Answers (Lesson 10-3)
(x Ϫ h)2 ϩ (y Ϫ k)2 ϭ r 2 r ϭ ͙(ෆx2 Ϫx1ෆ)2 ϩ (ෆy2 Ϫෆy1)2 Distance formula
r ϭ ͙ෆ(Ϫ4 Ϫෆ1)2 ϩෆ(5 Ϫ ෆ1)2 (x1, y1) ϭ (1, 1), (x2, y2) ϭ (Ϫ4, 5)
b. Write the equation of the circle with center (4, Ϫ3) and radius 5. Simplify.
ϭ ͙ෆ(Ϫ5)2ෆϩ 42 ϭ ͙ෆ41
(x Ϫ 4)2 ϩ (y ϩ 3)2 ϭ 25
A9 Glencoe Algebra 2 c. The circle with equation (x ϩ 8)2 ϩ y2 ϭ 121 has center (Ϫ8, 0) and radius The radius of the circle is ͙4ෆ1, so r2 ϭ 41.
11 . An equation of the circle is (x Ϫ 1)2 ϩ ( y Ϫ 1)2 ϭ 41.
d. The circle with equation (x Ϫ 10)2 ϩ ( y ϩ 10)2 ϭ 1 has center (10, Ϫ10) and Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 10-3
radius 1. Exercises
Write an equation for the circle that satisfies each set of conditions.
2. a. In order to find center and radius of the circle with equation x2 ϩ y2 ϩ 4x Ϫ 6y Ϫ3 ϭ 0,
1. center (8, Ϫ3), radius 6 (x Ϫ 8)2 ϩ (y ϩ 3)2 ϭ 36
it is necessary to complete the square . Fill in the missing parts of this
process.
2. center (5, Ϫ6), radius 4 (x Ϫ 5)2 ϩ (y ϩ 6)2 ϭ 16
x2 ϩ y2 ϩ 4x Ϫ 6y Ϫ 3 ϭ 0 3. center (Ϫ5, 2), passes through (Ϫ9, 6) (x ϩ 5)2 ϩ (y Ϫ 2)2 ϭ 32
x2 ϩ y2 ϩ 4x Ϫ 6y ϭ 3 4. endpoints of a diameter at (6, 6) and (10, 12) (x Ϫ 8)2 ϩ (y Ϫ 9)2 ϭ 13
x2 ϩ 4x ϩ 4 ϩ y2 Ϫ 6y ϩ 9 ϭ 3 ϩ 4 ϩ 9
5. center (3, 6), tangent to the x-axis (x Ϫ 3)2 ϩ (y Ϫ 6)2 ϭ 36
(x ϩ 2 )2 ϩ ( y Ϫ 3 )2 ϭ 16
b. This circle has radius 4 and center at (Ϫ2, 3) .
6. center (Ϫ4, Ϫ7), tangent to x ϭ 2 (x ϩ 4)2 ϩ (y ϩ 7)2 ϭ 36
Remember What You Learned 7. center at (Ϫ2, 8), tangent to y ϭ Ϫ4 (x ϩ 2)2 ϩ (y Ϫ 8)2 ϭ 144
3. How can the distance formula help you to remember the equation of a circle? 8. center (7, 7), passes through (12, 9) (x Ϫ 7)2 ϩ (y Ϫ 7)2 ϭ 29
Sample answer: Write the distance formula. Replace (x1, y1) with (h, k) 9. endpoints of a diameter are (Ϫ4, Ϫ2) and (8, 4) (x Ϫ 2)2 ϩ (y Ϫ 1)2 ϭ 45
and (x2, y2) with (x, y ). Replace d with r. Square both sides. Now you
have the equation of a circle.
10. endpoints of a diameter are (Ϫ4, 3) and (6, Ϫ8) (x Ϫ 1)2 ϩ (y ϩ 2.5)2 ϭ 55.25
Chapter 10 20 Glencoe Algebra 2 Chapter 10 21 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-3 Study Guide and Intervention (continued) 10-3 Skills Practice
Circles Circles
Graph Circles To graph a circle, write the given equation in the standard form of the Write an equation for the circle that satisfies each set of conditions.
equation of a circle, (x Ϫ h)2 ϩ (y Ϫ k)2 ϭ r2. 1. center (0, 5), radius 1 unit 2. center (5, 12), radius 8 units
Plot the center (h, k) of the circle. Then use r to calculate and plot the four points (h ϩ r, k), x2 ϩ (y Ϫ 5)2 ϭ 1 (x Ϫ 5)2 ϩ (y Ϫ 12)2 ϭ 64
(h Ϫ r, k), (h, k ϩ r), and (h, k Ϫ r), which are all points on the circle. Sketch the circle that
goes through those four points. 3. center (4, 0), radius 2 units 4. center (2, 2), radius 3 units
Example Find the center and radius of the circle (x Ϫ 4)2 ϩ y 2 ϭ 4 (x Ϫ 2)2 ϩ (y Ϫ 2)2 ϭ 9
whose equation is x2 ϩ 2x ϩ y2 ϩ 4y ϭ 11. Then graph y 5. center (4, Ϫ4), radius 4 units 6. center (Ϫ6, 4), radius 5 units
x2 ϩ 2x ϩ y2 ϩ 4y ϭ 11
the circle. (x Ϫ 4)2 ϩ (y ϩ 4)2 ϭ 16 (x ϩ 6)2 ϩ (y Ϫ 4)2 ϭ 25
Ox
x2 ϩ 2x ϩ y2 ϩ 4y ϭ 11 7. endpoints of a diameter at (Ϫ12, 0) and (12, 0) x 2 ϩ y 2 ϭ 144
x2 ϩ 2x ϩ ■ ϩ y2 ϩ 4y ϩ ■ ϭ 11 ϩ■
x2 ϩ 2x ϩ 1 ϩ y2 ϩ 4y ϩ 4 ϭ 11 ϩ 1 ϩ 4 8. endpoints of a diameter at (Ϫ4, 0) and (Ϫ4, Ϫ6) (x ϩ 4)2 ϩ (y ϩ 3)2 ϭ 9
(x ϩ 1)2 ϩ ( y ϩ 2)2 ϭ 16 9. center at (7, Ϫ3), passes through the origin (x Ϫ 7)2 ϩ (y ϩ 3)2 ϭ 58 Answers (Lesson 10-3)
Therefore, the circle has its center at (Ϫ1, Ϫ2) and a radius of 10. center at (Ϫ4, 4), passes through (Ϫ4, 1) (x ϩ 4)2 ϩ (y Ϫ 4)2 ϭ 9
͙1ෆ6 ϭ 4. Four points on the circle are (3, Ϫ2), (Ϫ5, Ϫ2), (Ϫ1, 2), 11. center at (Ϫ6, Ϫ5), tangent to y-axis (x ϩ 6)2 ϩ (y ϩ 5)2 ϭ 36
and (Ϫ1, Ϫ6).
Exercises 12. center at (5, 1), tangent to x-axis (x Ϫ 5)2 ϩ (y Ϫ 1)2 ϭ 1
A10 Find the center and radius of the circle with the given equation. Then graph the Find the center and radius of the circle with the given equation. Then graph the
circle. circle.
1. (x Ϫ 3)2 ϩ y2 ϭ 9 2. x2 ϩ (y ϩ 5)2 ϭ 4 3. (x Ϫ 1)2 ϩ (y ϩ 3)2 ϭ 9 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 13. x2 ϩ y2 ϭ 9Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.14. (x Ϫ 1)2 ϩ (y Ϫ 2)2 ϭ 415. (x ϩ 1)2 ϩ y2 ϭ 16
(3, 0), r ϭ 3 (0, Ϫ5), r ϭ 2 (1, Ϫ3), r ϭ 3 (0, 0), 3 units Lesson 10-3(1, 2), 2 units(Ϫ1, 0), 4 units
y y y y y y
Ox Ox
Ox
Ox Ox Ox
4. (x Ϫ 2)2 ϩ (y ϩ 4)2 ϭ 16 5. x2 ϩ y2 Ϫ 10x ϩ 8y ϩ 16 ϭ 0 6. x2 ϩ y2 Ϫ 4x ϩ 6y ϭ 12
(2, Ϫ4), r ϭ 4 (5, Ϫ4), r ϭ 5 (2, Ϫ3), r ϭ 5 16. x2 ϩ (y ϩ 3)2 ϭ 81 17. (x Ϫ 5)2 ϩ (y ϩ 8)2 ϭ 49 18. x2 ϩ y2 Ϫ 4y Ϫ 32 ϭ 0
y yy (0, Ϫ3), 9 units (5, Ϫ8), 7 units (0, 2), 6 units
Ox Ox y y y
Ox 12 8
O 4 8 12 x
6 –4 4
Glencoe Algebra 2 –12 –6 O x –8 –8 –4 O 4 8x
–6 6 12 –4
–12
–8
–12
Chapter 10 22 Glencoe Algebra 2 Chapter 10 23 Glencoe Algebra 2
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-3 Practice 10-3 Word Problem Practice
Circles Circles
Write an equation for the circle that satisfies each set of conditions. 1. RADAR A scout plane is equipped 4. POOLS The pool on an architectural
with radar. The boundary of the floor plan is given by the equation
1. center (Ϫ4, 2), radius 8 units 2. center (0, 0), radius 4 units radar’s range is given by the circle x2 ϩ 6x ϩ y2 ϩ 8y ϭ 0. What point on
(x Ϫ 4)2 ϩ (y Ϫ 6)2 ϭ 4900. Each unit the edge of the pool is farthest from the
(x ϩ 4)2 ϩ (y Ϫ 2)2 ϭ 64 x2 ϩ y2 ϭ 16 corresponds to one mile. What is the origin?
maximum distance that an object can be (؊6, ؊8)
3. center Ϫᎏ41 , Ϫ͙ෆ3 , radius 5͙2ෆ units 4. center (2.5, 4.2), radius 0.9 unit from the plane and still be detected by
its radar? TREASURE For Exercises 5 and 6, use
x ϩ ᎏ41 2 ϩ (y ϩ ͙ෆ3)2 ϭ 50 (x Ϫ 2.5)2 ϩ (y Ϫ 4.2)2 ϭ 0.81 the following information.
y
5. endpoints of a diameter at (Ϫ2, Ϫ9) and (0, Ϫ5) (x ϩ 1)2 ϩ (y ϩ 7)2 ϭ 5 Scout plane A mathematically inclined pirate decided
to hide the location of a treasure by
6. center at (Ϫ9, Ϫ12), passes through (Ϫ4, Ϫ5) (x ϩ 9)2 ϩ (y ϩ 12)2 ϭ 74 5 marking it as the center of a circle given
by an equation in non-standard form.
7. center at (Ϫ6, 5), tangent to x-axis (x ϩ 6)2 ϩ (y Ϫ 5)2 ϭ 25
Find the center and radius of the circle with the given equation. Then graph the O 5x Answers (Lesson 10-3)
circle.
70 mi
8. (x ϩ 3)2 ϩ y2 ϭ 16 9. 3x2 ϩ 3y2 ϭ 12 10. x2 ϩ y2 ϩ 2x ϩ 6y ϭ 26 y
2. STORAGE An engineer uses a 5
(Ϫ3, 0), 4 units (0, 0), 2 units (Ϫ1, Ϫ3), 6 units coordinate plane to show the layout
of a side view of a storage building.
y y y The y-axis represents a wall and the
x-axis represents the floor. A 10-meter
4 diameter cylinder rests on its side flush
against the wall. On the side view, the
A11 –8 –4 O 4 8x cylinder is represented by a circle in the Ϫ5 O 5x
–4 first quadrant that is tangent to both Ϫ5
Ox Ox axes. Each unit represents 1 meter.
–8 What is the equation of this circle?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.(x ؊ 5)2 ؉ (y ؊ 5)2 ؍25
11. (x Ϫ 1)2 ϩ y2 ϩ 4y ϭ 12 12. x2 Ϫ 6x ϩ y2 ϭ 0 13. x2 ϩ y2 ϩ 2x ϩ 6y ϭ Ϫ1
Lesson 10-33. FERRIS WHEEL The Texas Star, the
(1, Ϫ2), 4 units (3, 0), 3 units (Ϫ1, Ϫ3), 3 units largest Ferris wheel in North America, The secret circle can be represented by:
is located in Dallas, Texas. It weighs x2 ϩ y2 Ϫ 2x ϩ 14y ϩ 49 ϭ 0.
y y y 678,554 pounds and can hold 264 riders
in its 44 gondolas. The Texas Star 5. Rewrite the equation of the circle in
Ox has a diameter of 212 feet. Use the standard form.
rectangular coordinate system with
Ox the origin on the ground directly below (x ؊ 1)2 ؉ (y ؉ 7)2 ؍1
Ox the center of the wheel and write the
equation of the circle that models the 6. Draw the circle on the map. Where is the
WEATHER For Exercises 14 and 15, use the following information. Texas Star. treasure?
x 2 ؉ (y ؊ 106)2 ؍11,236
On average, the circular eye of a hurricane is about 15 miles in diameter. Gale winds can See circle on map at (1, ؊7); the
affect an area up to 300 miles from the storm’s center. In 2005, Hurricane Katrina devastated southwest corner of Meadow
southern Louisiana. A satellite photo of Katrina’s landfall showed the center of its eye on one Madness.
coordinate system could be approximated by the point (80, 26).
Glencoe Algebra 2
14. Write an equation to represent a possible boundary of Katrina’s eye.
(x Ϫ 80)2 ϩ (y Ϫ 26)2 ϭ 56.25
15. Write an equation to represent a possible boundary of the area affected by gale winds.
(x Ϫ 80)2 ϩ (y Ϫ 26)2 ϭ 90,000
Chapter 10 24 Glencoe Algebra 2 Chapter 10 25 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-3 Enrichment 10-3 Graphing Calculator Activity
Tangents to Circles Matrices and Equations of Circles
A line that intersects a circle in exactly one point is A graphing calculator can be used to write the equation of a circle in the
a tangent to the circle. In the diagram, line ᐉ is form x2 ϩ y2 ϩ Dx ϩ Ey ϩ F ϭ 0 given any three points on the circle.
tangent to the circle with equation x2 ϩ y2 ϭ 25 at
the point whose coordinates are (3, 4). ᐉ y Write the equation of the circle that passes
through the given points. Identify the center and
A line is tangent to a circle at a point P on the circle x2 ϩ y2 ϭ 25 5 (3, 4) Example
if and only if the line is perpendicular to the radius
from the center of the circle to point P. This fact radius of each circle.
enables you to find an equation of the tangent to a
circle at a point P if you know an equation for the a. A(5, 3), B(Ϫ2, 2), and C(Ϫ1, Ϫ5)
circle and the coordinates of P.
–5 O 5x Substitute each ordered pair for (x, y) in x2 ϩ y2 ϩ Dx ϩ Ey ϩ F ϭ 0 to form
–5
the a system of equations.
5D ϩ 3E ϩ F ϭ Ϫ34 Ϫ2D ϩ 2E ϩ F ϭ Ϫ8 ϪD Ϫ 5E ϩ F ϭ Ϫ26
Solve the system using a matrix equation to find D, E, and F. Replace the
coefficients in the expanded form. Then, complete the square to write the
equation in standard form to identify the center and radius. Answers (Lesson 10-3)
Keystrokes: 2nd [MATRX] ENTER 3 ENTER 3 ENTER 5
Exercises ENTER 3 ENTER 1 ENTER (–) 2 ENTER 2 ENTER 1 ENTER (–) 1
Use the diagram above to solve each problem. ENTER (–) 5 ENTER 1 ENTER 2nd [MATRX] [EDIT] 2
1. What is the slope of the radius to the point with coordinates (3, 4)? What is ENTER 3 ENTER 1 ENTER (–) 34 ENTER (–) 8 ENTER (–) 26 ENTER
the slope of the tangent to that point?
2nd [QUIT] 2nd [MATRX] ENTER x –1 ϫ 2nd [MATRX] 2
ᎏ43ᎏ, Ϫᎏ43ᎏ
A12 ENTER .
Thus, D ϭ Ϫ4, E ϭ 2, and F ϭ Ϫ20.
The expanded form is x2 ϩ y2 Ϫ 4x ϩ 2y Ϫ 20 ϭ 0.
After completing the square, the standard form is (x Ϫ 2)2 ϩ (y ϩ 1)2 ϭ 25.
The center is ( 2, Ϫ1), and the radius is 5.
2. Find an equation of the line ᐉ that is tangent to the circle at (3, 4). Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. A(Ϫ2, 3), B(6, Ϫ5), and C(0, 7)
y ϭ Ϫᎏ43ᎏx ϩ ᎏ24ᎏ5 Lesson 10-3
Find a system of equations. Then enter the equations into an augmented
3. If k is a real number between Ϫ5 and 5, how many points on the circle have matrix. Reduce the matrix to row reduced echelon form using the rref(
x-coordinate k? State the coordinates of these points in terms of k. command. The row reduced echelon form of an augmented matrix will
display the solution to the system.
two, (k, Ϯ͙2ෆ5 Ϫ ෆk2 )
Ϫ2D ϩ 3E + F ϭ Ϫ13 6D Ϫ 5E ϩ F ϭ Ϫ61 7E ϩ F ϭ Ϫ49
Keystrokes: Enter the system of equations as [A], a 3 ϫ 4 augmented
matrix. Then use the reduced row echelon form by pressing 2nd
[MATRX] ALPHA [B] 2nd [MATRX] ENTER ) ENTER .
4. Describe how you can find equations for the tangents to the points you named The solution is D = Ϫ10, E = Ϫ4, and F = Ϫ21. The expanded form
for Exercise 3. is x2 ϩ y2 Ϫ 10x Ϫ 4y Ϫ 21 = 0, standard form is (x Ϫ 5)2 ϩ (y Ϫ 2)2
Use the coordinates of (0, 0) and of one of the given points. Find the ϭ 50. The center is (5, 2) and the radius is 5͙ෆ2.
slope of the radius to that point. Use the slope of the radius to find what
the slope of the tangent must be. Use the slope of the tangent and the Exercises
coordinates of the point on the circle to find an equation for the tangent.
Glencoe Algebra 2 Write the equation of the circle that passes through the given
5. Find an equation for the tangent at (Ϫ3, 4). points. Identify the center and radius of each circle.
y ϭ ᎏ34ᎏx ϩ ᎏ2x5ᎏ 1. (0, Ϫ1), (Ϫ3, Ϫ2), and (Ϫ6, Ϫ1) 2. (7, Ϫ1), (11, Ϫ5), and (3, Ϫ5) 3. (Ϫ2, 7), (Ϫ9, 0), and (Ϫ10, Ϫ5)
x2 ϩ y2 ϩ 6x Ϫ 6y Ϫ 7 ϭ 0; x2 ϩ y2 Ϫ 14x ϩ 10y ϩ 58 ϭ 0; x2ϩy2Ϫ 6x ϩ 10y Ϫ 135 ϭ 0;
C(Ϫ3, 3), R ϭ 5 C(7, Ϫ5), R ϭ 4 C(3, Ϫ5), R ϭ 13
Chapter 10 26 Glencoe Algebra 2 Chapter 10 27 Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-4 Lesson Reading Guide 10-4 Study Guide and Intervention
Ellipses Ellipses
Get Ready for the Lesson Equations of Ellipses An ellipse is the set of all points in a plane such that the sum
Read the introduction to Lesson 10-4 in your textbook. of the distances from two given points in the plane, called the foci, is constant. An ellipse
Is the Earth always the same distance from the Sun? Explain your answer using the words has two axes of symmetry which contain the major and minor axes. In the table, the
lengths a, b, and c are related by the formula c2 ϭ a2 Ϫ b2.
circle and ellipse. No; if the Earth’s orbit were a circle, it would always be the
same distance from the Sun because every point on a circle is the same Standard Form of Equation ᎏ(x Ϫa2h)2 ϩ ᎏ(y Ϫb2k)2 ϭ 1 (y Ϫ k)2 (x Ϫ h)2
distance from the center. However, the Earth’s orbit is an ellipse, and the (h, k) ᎏa2 ϩ ᎏb2 ϭ 1
points on an ellipse are not all the same distance from the center. Center
Direction of Major Axis Horizontal (h, k)
Read the Lesson Foci (h ϩ c, k), (h Ϫ c, k ) Vertical
Length of Major Axis (h, k Ϫ c), (h, k ϩ c)
1. An ellipse is the set of all points in a plane such that the sum of the Length of Minor Axis 2a units 2a units
2b units 2b units
distances from two fixed points is constant . The two fixed points are called the
foci of the ellipse. Example Answers (Lesson 10-4)
x2 y2 Write an equation for the ellipse shown. y
ᎏ9 ᎏ4 F1
2. Consider the ellipse with equation ϩ ϭ 1. The length of the major axis is the distance between (Ϫ2, Ϫ2)
and (Ϫ2, 8). This distance is 10 units.
a. For this equation, a ϭ 3 and b ϭ 2 .
2a ϭ 10, so a ϭ 5
b. Write an equation that relates the values of a, b, and c. c 2 ϭ a 2 Ϫ b 2
The foci are located at (Ϫ2, 6) and (Ϫ2, 0), so c ϭ 3.
c. Find the value of c for this ellipse. ͙5ෆ b2 ϭ a2 Ϫ c2 F2
ϭ 25 Ϫ 9 O
A13 3. Consider the ellipses with equations ᎏ2y52 ϩ ᎏ1x62 ϭ 1 and ᎏx92 ϩ ᎏy42 ϭ 1. Complete the ϭ 16 x
following table to describe characteristics of their graphs.
The center of the ellipse is at (Ϫ2, 3), so h ϭ Ϫ2, k ϭ 3,
a2 ϭ 25, and b2 ϭ 16. The major axis is vertical.
An equation of the ellipse is ᎏ( y Ϫ253)2 ϩ ᎏ(x ϩ162)2 ϭ 1.
Standard Form of Equation ᎏ2y52 ϩ ᎏ1x62 ϭ 1 ᎏx92 ϩ ᎏy42 ϭ 1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Direction of Major Axis Exercises
vertical horizontal Lesson 10-4
horizontal vertical Write an equation for the ellipse that satisfies each set of conditions.
(0, 3), (0, Ϫ3)
Direction of Minor Axis (͙ෆ5, 0), (Ϫ͙5ෆ, 0) 1. endpoints of major axis at (Ϫ7, 2) and (5, 2), endpoints of minor axis at (Ϫ1, 0) and (Ϫ1, 4)
Foci
ᎏ(x ϩ361)2 ϩ ᎏ(y Ϫ4 2)2 ϭ 1
Length of Major Axis 10 units 6 units
2. major axis 8 units long and parallel to the x-axis, minor axis 2 units long, center at (Ϫ2, Ϫ5)
Length of Minor Axis 8 units 4 units
ᎏ(x ϩ162)2 ϩ (y ϩ 5)2 ϭ 1
Glencoe Algebra 2 Remember What You Learned
3. endpoints of major axis at (Ϫ8, 4) and (4, 4), foci at (Ϫ3, 4) and (Ϫ1, 4)
4. Some students have trouble remembering the two standard forms for the equation of an
ellipse. How can you remember which term comes first and where to place a and b in ᎏ(x ϩ362)2 ϩ ᎏ(y Ϫ354)2 ϭ 1
these equations? The x-axis is horizontal. If the major axis is horizontal, the 4. endpoints of major axis at (3, 2) and (3, Ϫ14), endpoints of minor axis at (Ϫ1, Ϫ6) and (7, Ϫ6)
first term is ᎏax22 . The y-axis is vertical. If the major axis is vertical, the
first term is ᎏya22 . a is always the larger of the numbers a and b. ᎏ(y ϩ646)2 ϩ ᎏ(x Ϫ163)2 ϭ 1
5. minor axis 6 units long and parallel to the x-axis, major axis 12 units long, center at (6, 1)
ᎏ(y Ϫ361)2 ϩ ᎏ(x Ϫ9 6)2 ϭ 1
Chapter 10 28 Glencoe Algebra 2 Chapter 10 29 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-4 Study Guide and Intervention (continued) 10-4 Skills Practice
Ellipses Ellipses
Graph Ellipses To graph an ellipse, if necessary, write the given equation in the Write an equation for each ellipse.
standard form of an equation for an ellipse. 1. y 2. (0, 5) y 3. (0, 5) y
ᎏ(x Ϫa2h)2 ϩ ᎏ( y Ϫb2k)2 ϭ 1 (for ellipse with major axis horizontal) or (–3, 0) (0, 2) (0, 3) (–4, 2) (4, 2)
ᎏ( y Ϫa2k)2 ϩ ᎏ(x Ϫb2h)2 ϭ 1 (for ellipse with major axis vertical) x
O (3, 0) x O x O
Use the center (h, k) and the endpoints of the axes to plot four points of the ellipse. To make (0, –2) (0, –3) (0, –1)
a more accurate graph, use a calculator to find some approximate values for x and y that
satisfy the equation.
Example Graph the ellipse 4x2 ϩ 6y2 ϩ 8x Ϫ 36y ϭ Ϫ34. (0, –5)
4x2 ϩ 6y2 ϩ 8x Ϫ 36y ϭ Ϫ34 y ᎏx92 ϩ ᎏy42 ϭ 1 ᎏ2y52 ϩ ᎏ1x62 ϭ 1 ᎏ1x62 ϩ ᎏ(y Ϫ9 2)2 ϭ 1 Answers (Lesson 10-4)
4x2 ϩ 8x ϩ 6y2 Ϫ 36y ϭ Ϫ 34 4x2 ϩ 6y2 ϩ 8x Ϫ 36y ϭ Ϫ34
4(x2 ϩ 2x ϩ ■) ϩ 6( y2 Ϫ 6y ϩ ■) ϭ Ϫ34 ϩ ■ Write an equation for the ellipse that satisfies each set of conditions.
4(x2 ϩ 2x ϩ 1) ϩ 6( y2 Ϫ 6y ϩ 9) ϭ Ϫ34 ϩ 58 Ox
4(x ϩ 1)2 ϩ 6( y Ϫ 3)2 ϭ 24 4. endpoints of major axis 5. endpoints of major axis 6. endpoints of major axis
ᎏ(x ϩ6 1)2 ϩ ᎏ( y Ϫ4 3)2 ϭ 1 at (0, 6) and (0, Ϫ6), at (2, 6) and (8, 6), at (7, 3) and (7, 9),
endpoints of minor axis
The center of the ellipse is (Ϫ1, 3). Since a2 ϭ 6, a ϭ ͙ෆ6. endpoints of minor axis endpoints of minor axis
Since b2 ϭ 4, b ϭ 2. at (Ϫ3, 0) and (3, 0)
A14 at (5, 4) and (5, 8) at (5, 6) and (9, 6)
The length of the major axis is 2͙6ෆ, and the length of the minor axis is 4. Since the x-term ᎏ3y62 ϩ ᎏx92 ϭ 1
has the greater denominator, the major axis is horizontal. Plot the endpoints of the axes. ᎏ(x Ϫ9 5)2 ϩ ᎏ(y Ϫ4 6)2 ϭ 1 ᎏ(y Ϫ9 6)2 ϩ ᎏ(x Ϫ4 7)2 ϭ 1
Then graph the ellipse. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 10-4
7. major axis 12 units long 8. endpoints of major axis 9. endpoints of major axis at
and parallel to x-axis,
Exercises minor axis 4 units long, at (Ϫ6, 0) and (6, 0), foci (0, 12) and (0, Ϫ12), foci at
center at (0, 0)
Find the coordinates of the center and the lengths of the major and minor axes at (Ϫ͙3ෆ2, 0) and (͙3ෆ2, 0) (0, ͙ෆ23 ) and (0, Ϫ͙ෆ23 )
for the ellipse with the given equation. Then graph the ellipse. ᎏ3x62 ϩ ᎏy42 ϭ 1
ᎏ3x62 ϩ ᎏy42 ϭ 1 ᎏ1y424 ϩ ᎏ1x221 ϭ 1
1. ᎏ1y22 ϩ ᎏx92 ϭ 1 (0, 0), 4͙3ෆ, 6 2. ᎏ2x52 ϩ ᎏy42 ϭ 1 (0, 0), 10, 4
yy Find the coordinates of the center and foci and the lengths of the major and
minor axes for the ellipse with the given equation. Then graph the ellipse.
Ox Ox 10. ᎏ1y020 ϩ ᎏ8x12 ϭ 1 11. ᎏ8x12 ϩ ᎏy92 ϭ 1 12. ᎏ4y92 ϩ ᎏ2x52 ϭ 1
(0, 0); (0, Ϯ͙1ෆ9); (0, 0); (Ϯ6͙ෆ2, 0); (0, 0), (0, Ϯ2͙ෆ6);
20; 18 18; 6 14; 10
3. x2 ϩ 4y2 ϩ 24y ϭ Ϫ32 (0, Ϫ3), 4, 2 4. 9x2 ϩ 6y2 Ϫ 36x ϩ 12y ϭ 12 (2, Ϫ1), 6, 2͙ෆ6 yyy
888
yy
444
Glencoe Algebra 2 Ox Ox –8 –4 O 4 8x –8 –4 O 4 8x –8 –4 O 4 8x
–4 –4 –4
–8 –8 –8
Chapter 10 30 Glencoe Algebra 2 Chapter 10 31 Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-4 Practice 10-4 Word Problem Practice
Ellipses Ellipses
Write an equation for each ellipse. 1. PERSPECTIVE A graphic designer uses 4. ASTRONOMY The orbit of an asteroid
an ellipse to draw a circle from the is given by the equation ᎏ4x020 ϩ ᎏ4y421 ϭ 1,
1. (0, 3) y 2. (0, 2 ϩ ͙5ළ) (y0, 5) 3. y horizontal perspective. The equation
(4, 3) used is ᎏ2x52 ϩ y2 ϭ 1. Graph this ellipse. where each unit represents one
2 (–6, 3)
y astronomical unit (i.e. the distance from
(–11, 0) 6 (11, 0) (–5, 3) (3, 3)
–12 –6 O 12 x Sun to Earth). What are the lengths of
–2 O x the major and minor axes of the orbit?
(0, 2 Ϫ ͙5ළ) (0, –1)
(0, –3) Ox Major axis: 42 astronomical units,
Minor axis: 40 astronomical units
ᎏ1x221 ϩ ᎏy92 ϭ 1 ᎏ(y Ϫ9 2)2 ϩ ᎏx42 ϭ 1 ᎏ(x ϩ251)2 ϩ ᎏ(y Ϫ9 3)2 ϭ 1
Ϫ5 O 5x
Write an equation for the ellipse that satisfies each set of conditions.
4. endpoints of major axis 5. endpoints of major axis 6. major axis 20 units long 2. ECHOES The walls of an elliptical room MODELING For Exercises 5 and 6, use Answers (Lesson 10-4)
and parallel to x-axis, are given by the equation ᎏ2x52 ϩ ᎏ1y62 ϭ 1. the following information.
at (Ϫ9, 0) and (9, 0), at (4, 2) and (4, Ϫ8), minor axis 10 units long, Two people want to stand at the foci
endpoints of minor axis endpoints of minor axis center at (2, 1) of the ellipse so that they can whisper James wants to try to make an ellipse using
to each other without anybody else a piece of string 26 inches long. He tacks
at (0, 3) and (0, Ϫ3) at (1, Ϫ3) and (7, Ϫ3) ᎏ(x1Ϫ002)2 ϩ ᎏ(y Ϫ251)2 ϭ 1 hearing. What are the coordinates of the two ends down 10 inches apart. He
the foci? then takes a pen and pulls the string taut.
ᎏ8x12 ϩ ᎏy92 ϭ 1 ᎏ(y ϩ253)2 ϩ ᎏ(x Ϫ9 4)2 ϭ 1 (3, 0) and (؊3, 0) He keeps the string taut and pulls the pen
around the tacks. By doing this, he creates
A15 7. major axis 10 units long, 8. major axis 16 units long, 9. endpoints of minor axis 3. FLASHLIGHTS Daniella ended up doing an ellipse.
minor axis 6 units long center at (0, 0), foci at at (0, 2) and (0, Ϫ2), foci her math homework late at night. To
and parallel to x-axis, at (Ϫ4, 0) and (4, 0) avoid disturbing others, she worked in 5. Determine the lengths of the major and
center at (2, Ϫ4) (0, 2͙1ෆ5 ) and (0, Ϫ2͙ෆ15 ) bed with a pen light. One problem asked minor axes of the ellipse that James
ᎏ2x02 ϩ ᎏy42 ϭ 1 her to draw an ellipse. She noticed that drew.
ᎏ(y ϩ254)2 ϩ ᎏ(x Ϫ9 2)2 ϭ 1 ᎏ6y42 ϩ ᎏx42 ϭ 1 her pen light created an elliptical patch
of light on her paper, so she simply Major axis has length 26, minor
traced the outline of the patch of light. axis has length 24.
The outline of the ellipse is shown below.
What is the equation of this ellipse in 6. If a coordinate grid is overlaid on the
standard form? ellipse so that the tacks are located at
(5, 0) and (Ϫ5, 0), what is the equation
y of the ellipse in standard form?
x ᎏ1x629 ؉ ᎏ1y424 ؍1
Find the coordinates of the center and foci and the lengths of the major and Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
minor axes for the ellipse with the given equation. Then graph the ellipse. Lesson 10-4
10. ᎏ1y62 ϩ ᎏx92 ϭ 1 11. ᎏ( y Ϫ361)2 ϩ ᎏ(x Ϫ1 3)2 ϭ 1 12. ᎏ(x ϩ494)2 ϩ ᎏ( y ϩ253)2 ϭ 1
(0, 0); (0, Ϯ͙ෆ7); 8; 6 (3, 1); (3, 1 Ϯ ͙3ෆ5 ); (Ϫ4, Ϫ3);
12; 2 (Ϫ4 Ϯ 2͙ෆ6, Ϫ3); 14; 10
yy y
8 4
4 –8 –4 O 4x
–4
Ox –8 –4 O 4 8x
–4 –8
–8 –12
Glencoe Algebra 2 13. SPORTS An ice skater traces two congruent ellipses to form a figure eight. Assume that the ᎏ4x92 ؉ ᎏ3y62 ؍1
center of the first loop is at the origin, with the second loop to its right. Write an equation
Chapter 10
to model the first loop if its major axis (along the x-axis) is 12 feet long and its minor
axis is 6 feet long. Write another equation to model the second loop.
ᎏ3x62 ϩ ᎏy92 ϭ 1; ᎏ(x Ϫ36ᎏ12)2 ϩ ᎏy92 ϭ 1
Chapter 10 32 Glencoe Algebra 2 33 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-4 Enrichment 10-5 Lesson Reading Guide
Eccentricity Hyperbolas
In an ellipse, the ratio ᎏacᎏ is called the eccentricity and is denoted by the Get Ready for the Lesson
letter e. Eccentricity measures the elongation of an ellipse. The closer e is to 0, Read the introduction to Lesson 10-5 in your textbook.
the more an ellipse looks like a circle. The closer e is to 1, the more elongated Look at the sketch of a hyperbola in the introduction to this lesson. List three ways in which
hyperbolas are different from parabolas.
it is. Recall that the equation of an ellipse is ᎏaxᎏ22 ϩ y2 ϭ 1 or ᎏbxᎏ22 ϩ y2 ϭ 1
ᎏbᎏ2 ᎏaᎏ2 Sample answer: A hyperbola has two branches, while a parabola is one
continuous curve. A hyperbola has two foci, while a parabola has one
where a is the length of the major axis, and that c ϭ ͙aෆ2 Ϫ bෆ2. focus. A hyperbola has two vertices, while a parabola has one vertex.
Read the Lesson Answers (Lessons 10-4 and 10-5)
Find the eccentricity of each ellipse rounded to the nearest 1. The graph at the right shows the hyperbola whose y
equation in standard form is ᎏ1x62 Ϫ ᎏy92 ϭ 1.
hundredth. y ϭ – 43x y ϭ 43x
1. ᎏx9ᎏ2 ϩ y2 ϭ 1 2. ᎏ8xᎏ12 ϩ y2 ϭ 1 3. ᎏx4ᎏ2 ϩ y2 ϭ 1 (–5, 0) (5, 0)
ᎏ3ᎏ6 ᎏ9ᎏ ᎏ9ᎏ (–4, 0) O
center
0.87 0.94 0.75 The point (0, 0) is the of the (4, 0) x
hyperbola.
vertices
4. ᎏ1xᎏ62 ϩ ᎏy9ᎏ2 ϭ 1 5. ᎏ3xᎏ62 ϩ ᎏ1y6ᎏ2 ϭ 1 6. ᎏx4ᎏ2 ϩ ᎏ3yᎏ62 ϭ 1 The points (4, 0) and (Ϫ4, 0) are the foci
of the hyperbola.
0.66 0.75 0.94
A16 The points (5, 0) and (Ϫ5, 0) are the
of the hyperbola.
The segment connecting (4, 0) and (Ϫ4, 0) is called the transverse axis.
conjugate axis.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The segment connecting (0, 3) and (0, Ϫ3) is called the
7. Is a circle an ellipse? Explain your reasoning. Lesson 10-5
The lines y ϭ ᎏ43 x and y ϭ Ϫᎏ43 x are called the asymptotes .
Yes; it is an ellipse with eccentricity 0.
8. The center of the sun is one focus of Earth's orbit around the sun. The 2. Study the hyperbola graphed at the right. y
length of the major axis is 186,000,000 miles, and the foci are 3,200,000 Ox
miles apart. Find the eccentricity of Earth's orbit. The center is (0, 0) .
The value of a is 2 .
approximately 0.017 The value of c is 4 .
To find b2, solve the equation c2 ϭ a2 ϩ b2 .
The equation in standard form for this hyperbola is ᎏx4ᎏ2 Ϫ ᎏ1y2ᎏ2 ϭ 1 .
Glencoe Algebra 2 9. An artificial satellite orbiting the earth travels at an altitude that varies Remember What You Learned
between 132 miles and 583 miles above the surface of the earth. If the
center of the earth is one focus of its elliptical orbit and the radius of the 3. What is an easy way to remember the equation relating the values of a, b, and c for a
earth is 3950 miles, what is the eccentricity of the orbit?
hyperbola? This equation looks just like the Pythagorean Theorem,
approximately 0.052 although the variables represent different lengths in a hyperbola than in
a right triangle.
Chapter 10 34 Glencoe Algebra 2 Chapter 10 35 Glencoe Algebra 2
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-5 Study Guide and Intervention 10-5 Study Guide and Intervention (continued)
Hyperbolas Hyperbolas
Equations of Hyperbolas A hyperbola is the set of all points in a plane such that Graph Hyperbolas To graph a hyperbola, write the given equation in the standard
the absolute value of the difference of the distances from any point on the hyperbola to any form of an equation for a hyperbola
two given points in the plane, called the foci, is constant.
ᎏ(x Ϫa2h)2 Ϫ ᎏ( y Ϫbᎏ2k)2 ϭ 1 if the branches of the hyperbola open left and right, or
In the table, the lengths a, b, and c are related by the formula c2 ϭ a2 ϩ b2. ᎏ( y Ϫa2k)2 Ϫ ᎏ(x Ϫb2h)2 ϭ 1 if the branches of the hyperbola open up and down
Standard Form of Equation ᎏ(x Ϫa2h)2 Ϫ ᎏ(y Ϫb2k)2 ϭ 1 ᎏ(y Ϫa2k)2 Ϫ ᎏ(x Ϫb2h)2 ϭ 1 Graph the point (h, k), which is the center of the hyperbola. Draw a rectangle with
y Ϫ k ϭ Ϯᎏab (x Ϫ h) y Ϫ k ϭ Ϯᎏab (x Ϫ h) dimensions 2a and 2b and center (h, k). If the hyperbola opens left and right, the vertices
Equations of the Asymptotes Horizontal Vertical are (h Ϫ a, k) and (h ϩ a, k). If the hyperbola opens up and down, the vertices are (h, k Ϫ a)
Transverse Axis (h Ϫ c, k), (h ϩ c, k) (h, k Ϫ c), (h, k ϩ c) and (h, k ϩ a).
Foci (h Ϫ a, k), (h ϩ a, k) (h, k Ϫ a), (h, k ϩ a)
Vertices Example Draw the graph of 6y2 Ϫ 4x2 Ϫ 36y Ϫ 8x ϭ Ϫ26.
Example Write an equation for the hyperbola with vertices (Ϫ2, 1) and (6, 1) Complete the squares to get the equation in standard form. y Answers (Lesson 10-5)
Ox
and foci (Ϫ4, 1) and (8, 1). 6y2 Ϫ 4x2 Ϫ 36y Ϫ 8x ϭ Ϫ26
6( y2 Ϫ 6y ϩ ■) Ϫ 4(x2 ϩ 2x ϩ ■) ϭ Ϫ26 ϩ ■
Use a sketch to orient the hyperbola correctly. The center of y 6( y2 Ϫ 6y ϩ 9) Ϫ 4(x2 ϩ 2x ϩ 1) ϭ Ϫ26 ϩ 50
the hyperbola is the midpoint of the segment joining the two O 6( y Ϫ 3)2 Ϫ 4(x ϩ 1)2 ϭ 24
vertices. The center is (ᎏϪ22ϩ 6 , 1), or (2, 1). The value of a is the ᎏ( y Ϫ4 3)2 Ϫ ᎏ(x ϩ6 1)2 ϭ 1
A17 distance from the center to a vertex, so a ϭ 4. The value of c is x The center of the hyperbola is (Ϫ1, 3).
the distance from the center to a focus, so c ϭ 6. According to the equation, a2 ϭ 4 and b2 ϭ 6, so a ϭ 2 and b ϭ ͙6ෆ.
c2 ϭ a2 ϩ b2 The transverse axis is vertical, so the vertices are (Ϫ1, 5) and (Ϫ1, 1). Draw a rectangle with
62 ϭ 42 ϩ b2 vertical dimension 4 and horizontal dimension 2͙6ෆ Ϸ 4.9. The diagonals of this rectangle
b2 ϭ 36 Ϫ 16 ϭ 20 are the asymptotes. The branches of the hyperbola open up and down. Use the vertices and
the asymptotes to sketch the hyperbola.
Use h, k, a2, and b2 to write an equation of the hyperbola. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
ᎏ(x Ϫ162)2 Ϫ ᎏ( y Ϫ201)2 ϭ 1 Exercises
Lesson 10-5
Exercises Find the coordinates of the vertices and foci and the equations of the asymptotes
Write an equation for the hyperbola that satisfies each set of conditions. for the hyperbola with the given equation. Then graph the hyperbola.
1. vertices (Ϫ7, 0) and (7, 0), conjugate axis of length 10 ᎏ4x92 Ϫ ᎏ2y52 ϭ 1 1. x2 Ϫ y2 ϭ 1 2. ( y Ϫ 3)2 Ϫ ᎏ(x ϩ9 2)2 ϭ 1 3. y2 Ϫ x2 ϭ 1
2. vertices (Ϫ2, Ϫ3) and (4, Ϫ3), foci (Ϫ5, Ϫ3) and (7, Ϫ3) ᎏ(x Ϫ9 1)2 Ϫ ᎏ(y ϩ273)2 ϭ 1 ᎏ4 ᎏ16 ᎏ16 ᎏ9
3. vertices (4, 3) and (4, Ϫ5), conjugate axis of length 4 ᎏ(y ϩ161)2 Ϫ ᎏ(x Ϫ4 4)2 ϭ 1 (Ϫ2, 4), (Ϫ2, 2);
4. vertices (Ϫ8, 0) and (8, 0), equation of asymptotes y ϭ Ϯᎏ61 x ᎏ6x42 Ϫ ᎏ91y62 ϭ 1 (2, 0), (Ϫ2, 0); (0, 4), (0, Ϫ4);
5. vertices (Ϫ4, 6) and (Ϫ4, Ϫ2), foci (Ϫ4, 10) and (Ϫ4, Ϫ6) ᎏ(y Ϫ162)2 Ϫ ᎏ(x ϩ484)2 ϭ 1 (Ϫ2, 3 ϩ ͙1ෆ0 ),
(2͙ෆ5, 0), (Ϫ2͙ෆ5, 0); (Ϫ2, 3 Ϫ ͙1ෆ0 ); (0, 5), (0, Ϫ5);
y ϭ Ϯᎏ34 x
y ϭ Ϯ2x y ϭ ᎏ13 x ϩ 3ᎏ32 ,
y ϭ Ϫᎏ13 x ϩ 2ᎏ13
y y
y
Glencoe Algebra 2 Ox Ox
O x
Chapter 10 36 Glencoe Algebra 2 Chapter 10 37 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-5 Skills Practice 10-5 Practice
Hyperbolas Hyperbolas
Write an equation for each hyperbola. Write an equation for each hyperbola.
1. y 2. y 3. y 1. y 2. y 3. y
8 (0, 3͙5ළ)
8 8 8 –8 4 (–3, 2 ϩ ͙ෆ34)8
(0, ͙6ෆ1) 4 (0, 6)
(–5, 0) 4 4 (2, 0) (0, 3) (–3, 5) 4 O x
(–2, 0) (–1, –2) (3, –2)
(5, 0) 4 8x –4 O 4 8 x
–8 –4 O 4 8x –8 –4 O 4 8x –8 –4 O (͙2ෆ9, 0) (–3, –1) O
(0, –͙ෆ61) –4 (0, –6) (–͙2ෆ9, 0) –4 –4 (0, –3) –8 –4 4x
–4 –8 (0, –3͙5ළ)
(–͙4ෆ1, 0) (͙4ෆ1, 0) –8 –4 (1, –2)
(–3, 2 Ϫ ͙ෆ34)
–8 –8
ᎏ2x52 Ϫ ᎏ1y62 ϭ 1 ᎏ3y62 Ϫ ᎏ2x52 ϭ 1 ᎏx42 Ϫ ᎏ2y52 ϭ 1 ᎏy92 Ϫ ᎏ3x62 ϭ 1 ᎏ(y Ϫ9 2)2 Ϫ ᎏ(x ϩ253)2 ϭ 1 ᎏ(x Ϫ4 1)2 Ϫ ᎏ(y ϩ162)2 ϭ 1
Write an equation for the hyperbola that satisfies each set of conditions. Write an equation for the hyperbola that satisfies each set of conditions. Answers (Lesson 10-5)
4. vertices (Ϫ4, 0) and (4, 0), conjugate axis of length 8 ᎏ1x62 Ϫ ᎏ1y62 ϭ 1 4. vertices (0, 7) and (0, Ϫ7), conjugate axis of length 18 units ᎏ4y92 Ϫ ᎏ8x12 ϭ 1
5. vertices (0, 6) and (0, Ϫ6), conjugate axis of length 14 ᎏ3y62 Ϫ ᎏ4x92 ϭ 1
A18 6. vertices (0, 3) and (0, Ϫ3), conjugate axis of length 10 ᎏy92 Ϫ ᎏ2x52 ϭ 1 5. vertices (1, Ϫ1) and (1, Ϫ9), conjugate axis of length 6 units ᎏ(y ϩ165)2 Ϫ ᎏ(x Ϫ9 1)2 ϭ 1
7. vertices (Ϫ2, 0) and (2, 0), conjugate axis of length 4 ᎏx42 Ϫ ᎏy42 ϭ 1 6. vertices (Ϫ5, 0) and (5, 0), foci (Ϯ͙2ෆ6, 0) ᎏ2x52 Ϫ ᎏy12 ϭ 1
8. vertices (Ϫ3, 0) and (3, 0), foci (Ϯ5, 0) ᎏx92 Ϫ ᎏ1y62 ϭ 1 7. vertices (1, 1) and (1, Ϫ3), foci (1, Ϫ1 Ϯ ͙5ෆ) ᎏ(y ϩ4 1)2 Ϫ ᎏ(x Ϫ1 1)2 ϭ 1
9. vertices (0, 2) and (0, Ϫ2), foci (0, Ϯ3) ᎏy42 Ϫ ᎏx52 ϭ 1
10. vertices (0, Ϫ2) and (6, Ϫ2), foci (3 Ϯ ͙1ෆ3, Ϫ2) ᎏ(x Ϫ9 3)2 Ϫ ᎏ(y ϩ4 2)2 ϭ 1 Find the coordinates of the vertices and foci and the equations of the asymptotes
for the hyperbola with the given equation. Then graph the hyperbola.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8. y2 Ϫ x2 ϭ 1 9. ᎏ( y Ϫ1 2)2 Ϫ ᎏ(x Ϫ4 1)2 ϭ 1 10. ᎏ( y ϩ4 2)2 Ϫ ᎏ(x Ϫ4 3)2 ϭ 1
Lesson 10-5ᎏ16ᎏ4
(1, 3), (1, 1); (3, 0), (3, Ϫ4);
(0, Ϯ4); (0, Ϯ2͙ෆ5);
y ϭ Ϯ2x (1, 2 Ϯ ͙ෆ5); (3, Ϫ2 Ϯ 2͙ෆ2);
Find the coordinates of the vertices and foci and the equations of the asymptotes y Ϫ 2 ϭ Ϯᎏ12 (x Ϫ 1) y ϩ 2 ϭ Ϯ(x Ϫ 3)
for the hyperbola with the given equation. Then graph the hyperbola.
y yy
11. ᎏx92 Ϫ ᎏ3y62 ϭ 1 12. ᎏ4y92 Ϫ ᎏx92 ϭ 1 13. ᎏ1x62 Ϫ ᎏy12 ϭ 1 8
(Ϯ3, 0); (Ϯ3͙5ෆ, 0); (0, Ϯ7); (0, Ϯ͙ෆ58 ); (Ϯ4, 0); (Ϯ͙1ෆ7, 0); 4 Ox
y ϭ Ϯ2x y ϭ Ϯᎏ37 x y ϭ Ϯᎏ14 x –8 –4 O 4 8x
–4
Ox
–8
yyy
88
44 11. ASTRONOMY Astronomers use special X-ray telescopes to observe the sources of
celestial X rays. Some X-ray telescopes are fitted with a metal mirror in the shape of a
Glencoe Algebra 2 Ox –8 –4 O 4 8x –8 –4 O 4 8x hyperbola, which reflects the X rays to a focus. Suppose the vertices of such a mirror are
–4 –4 located at (Ϫ3, 0) and (3, 0), and one focus is located at (5, 0). Write an equation that
models the hyperbola formed by the mirror. ᎏx92 Ϫ ᎏ1y62 ϭ 1
–8 –8
Chapter 10 38 Glencoe Algebra 2 Chapter 10 39 Glencoe Algebra 2
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-5 Word Problem Practice 10-5 Enrichment
Hyperbolas 4. ASTRONOMY Astronomers discover Rectangular Hyperbolas
a new comet. They study its path and
1. LIGHTHOUSES The location of a discover that it can be modeled by a A rectangular hyperbola is a hyperbola with perpendicular asymptotes.
lighthouse is represented by the origin branch of a hyperbola with equation For example, the graph of x2 Ϫ y2 ϭ 1 is a rectangular hyperbola. A hyperbola
of a coordinate plane. A boat in the 4x2 Ϫ 40x Ϫ 25y2 ϭ 0. Rewrite this with asymptotes that are not perpendicular is called a nonrectangular
distance appears to be on a collision equation in standard form and find the hyperbola. The graphs of equations of the form xy ϭ c, where c is a constant,
course with the lighthouse. However, center of the hyperbola. are rectangular hyperbolas.
the boat veers off and turns away at ᎏ(x ؊255)2 ؊ ᎏy42 ؍1, center at (5, 0)
the last moment, avoiding the rocky Make a table of values and plot points to graph each rectangular
shallows. The path followed by the boat LIGHTNING For Exercises 5–7, use the
is modeled by a branch of the hyperbola following information. hyperbola below. Be sure to consider negative values for the
with equation ᎏ9x020 Ϫ ᎏ4y020 ϭ 1. If the
unit length corresponds to a yard, Brittany and Kirk were talking on the variables. See students’ tables.
how close did the boat come to the phone when Brittany heard the thunder
lighthouse? from a lightning bolt outside. Eight seconds 1. xy ϭ Ϫ4 2. xy ϭ 3 Answers (Lesson 10-5)
30 yd later, she could hear the same thunder over
the phone. Brittany and Kirk live 2 miles y y
2. FIND THE ERROR Curtis was trying apart and sounds travels about 1 mile every
to write the equation for a hyperbola 5 seconds. Ox Ox
with a vertical transverse axis of length
A19 10 and conjugate axis of length 6. The 5. On a coordinate plane, assume that Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
equation he got was ᎏy92 Ϫ ᎏ2x52 ϭ 1. Did Brittany is located at (Ϫ1, 0) and Kirk is
he make a mistake? If so, what did he located at (1, 0). Write an equation using 3. xy ϭ Ϫ1 Lesson 10-54. xy ϭ 8
do wrong? the Distance Formula that describes the
Yes. Curtis exchanged the possible locations of the lightning strike. y y
transverse and conjugate axes.
͙ෆ(x ؊ 1)ෆ2 ؉ y 2ෆ ؊͙ෆ(x ؉ 1ෆ)2 ؉ y ෆ2 ؍ᎏ58 Ox Ox
3. MIRROR At a carnival, designers are
planning a funhouse. They plan to put 6. Rewrite the equation you wrote for
a large hyperbolic mirror inside this Exercise 5 so it is in the standard form
funhouse. They design the mirror’s for a hyperbola.
hyperbolic cross section on graph paper
using a hyperbola with a horizontal ᎏxᎏ54ᎏ22 ؊ ᎏyᎏ35ᎏ22 ؍1
transverse axis. The asymptotes are to
be y ϭ 9x and y ϭ Ϫ9x so the mirror is 7. Which branch of the hyperbola
somewhat shallow. They also want the corresponds to the places where the
vertices to be 1 unit from the origin. lightning bolt might have struck?
What equation should they use for the
hyperbola? the left branch, the branch with
negative x coordinates
ᎏx12 ؊ ᎏy12 ؍1
Glencoe Algebra 2 5. Make a conjecture about the asymptotes of rectangular hyperbolas.
The coordinate axes are the asymptotes.
Chapter 10 40 Glencoe Algebra 2 Chapter 10 41 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-6 Lesson Reading Guide 10-6 Study Guide and Intervention
Conic Sections Conic Sections
Get Ready for the Lesson Standard Form Any conic section in the coordinate plane can be described by an
Read the introduction to Lesson 10-6 in your textbook. equation of the form
Ax2 ϩ Bxy ϩ Cy2 ϩ Dx ϩ Ey ϩ F ϭ 0, where A, B, and C are not all zero.
The figures in the introduction show how a plane can slice a double cone to form the conic
sections. Name the conic section that is formed if the plane slices the double cone in each of One way to tell what kind of conic section an equation represents is to rearrange terms and
the following ways: complete the square, if necessary, to get one of the standard forms from an earlier lesson.
This method is especially useful if you are going to graph the equation.
• The plane is parallel to the base of the double cone and slices through one of the cones Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example Write the equation 3x2 Ϫ 4y2 Ϫ 30x Ϫ 8y ϩ 59 ϭ 0 in standard form.
that form the double cone. circle Lesson 10-6
State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.
• The plane is perpendicular to the base of the double cone and slices through both of the
3x2 Ϫ 4y2 Ϫ 30x Ϫ 8y ϩ 59 ϭ 0 Original equation
cones that form the double cone. hyperbola 3x2 Ϫ 30x Ϫ 4y2 Ϫ 8y ϭ Ϫ59 Isolate terms.
Factor out common multiples.
Read the Lesson 3(x2 Ϫ 10x ϩ ■) Ϫ 4( y2 ϩ 2y ϩ ■) ϭ Ϫ59 ϩ ■ ϩ ■ Complete the squares. Answers (Lesson 10-6)
3(x2 Ϫ 10x ϩ 25) Ϫ 4( y2 ϩ 2y ϩ 1) ϭ Ϫ59 ϩ 3(25) ϩ (Ϫ4)(1)
1. Name the conic section that is the graph of each of the following equations. Give the Simplify.
coordinates of the vertex if the conic section is a parabola and of the center if it is a 3(x Ϫ 5)2 Ϫ 4( y ϩ 1)2 ϭ 12
circle, an ellipse, or a hyperbola. Divide each side by 12.
ᎏ(x Ϫ4 5)2 Ϫ ᎏ( y ϩ3 1)2 ϭ 1
a. ᎏ(x Ϫ363)2 ϩ ᎏ( y ϩ155)2 ϭ 1 ellipse; (3, Ϫ5)
The graph of the equation is a hyperbola with its center at (5, Ϫ1). The length of the
b. x ϭ Ϫ2( y ϩ 1)2 ϩ 7 parabola; (7, Ϫ1) transverse axis is 4 units and the length of the conjugate axis is 2͙ෆ3 units.
c. (x Ϫ 5)2 Ϫ ( y ϩ 5)2 ϭ 1 hyperbola; (5, Ϫ5) Exercises
A20 d. (x ϩ 6)2 ϩ ( y Ϫ 2)2 ϭ 1 circle; (Ϫ6, 2) Write each equation in standard form. State whether the graph of the equation is
a parabola, circle, ellipse, or hyperbola.
2. Each of the following is the equation of a conic section. For each equation, identify the Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. x2 ϩ y2 Ϫ 6x ϩ 4y ϩ 3 ϭ 0 2. x2 ϩ 2y2 ϩ 6x Ϫ 20y ϩ 53 ϭ 0
values of A and C. Then, without writing the equation in standard form, state whether
the graph of each equation is a parabola, circle, ellipse, or hyperbola. (x Ϫ 3)2 ϩ (y ϩ 2)2 ϭ 10; circle ᎏ(x ϩ6 3)2 ϩ ᎏ(y Ϫ3 5)2 ϭ 1; ellipse
a. 2x2 ϩ y2 Ϫ 6x ϩ 8y ϩ 12 ϭ 0 A ϭ 2 ; C ϭ 1 ; type of graph: ellipse 3. 6x2 Ϫ 60x Ϫ y ϩ 161 ϭ 0 4. x2 ϩ y2 Ϫ 4x Ϫ14y ϩ 29 ϭ 0
y ϭ 6(x Ϫ 5)2 ϩ 11; parabola (x Ϫ 2)2 ϩ (y Ϫ 7)2 ϭ 24; circle
b. 2x2 ϩ 3x Ϫ 2y Ϫ 5 ϭ 0 A ϭ 2 ; C ϭ 0 ; type of graph: parabola 5. 6x2 Ϫ 5y2 ϩ 24x ϩ 20y Ϫ 56 ϭ 0 6. 3y2 ϩ x Ϫ 24y ϩ 46 ϭ 0
c. 5x2 ϩ 10x ϩ 5y2 Ϫ 20y ϩ 1 ϭ 0 A ϭ 5 ; C ϭ 5 ; type of graph: circle ᎏ(x ϩ102)2 Ϫ ᎏ(y Ϫ122)2 ϭ 1; hyperbola x ϭ Ϫ3(y Ϫ 4)2 ϩ 2; parabola
d. x2 Ϫ y2 ϩ 4x ϩ 2y Ϫ 5 ϭ 0 A ϭ 1 ; C ϭ Ϫ1 ; type of graph: hyperbola 7. x2 Ϫ 4y2 Ϫ 16x ϩ 24y Ϫ 36 ϭ 0 8. x2 ϩ 2y2 ϩ 8x ϩ 4y ϩ 2 ϭ 0
Remember What You Learned ᎏ(x Ϫ648)2 Ϫ ᎏ(y Ϫ163)2 ϭ 1; hyperbola ᎏ(x ϩ164)2 ϩ ᎏ(y ϩ8 1)2 ϭ 1; ellipse
3. What is an easy way to recognize that an equation represents a parabola rather than 9. 4x2 ϩ 48x ϩ y ϩ 158 ϭ 0 10. 3x2 ϩ y2 Ϫ 48x Ϫ 4y ϩ 184 ϭ 0
one of the other conic sections?
y ϭ Ϫ4(x ϩ 6)2 Ϫ 14; parabola ᎏ(x Ϫ4 8)2 ϩ ᎏ(y Ϫ122)2 ϭ 1; ellipse
If the equation has an x2 term and y term but no y2 term, then the graph
Glencoe Algebra 2 is a parabola. Likewise, if the equation has a y2 term and x term but no 11. Ϫ3x2 ϩ 2y2 Ϫ 18x ϩ 20y ϩ 5 ϭ 0 12. x2 ϩ y2 ϩ 8x ϩ 2y ϩ 8 ϭ 0
x2 term, then the graph is a parabola.
ᎏ(y ϩ9 5)2 Ϫ ᎏ(x ϩ6 3)2 ϭ 1; hyperbola (x ϩ 4)2 ϩ (y ϩ 1)2 ϭ 9; circle
Chapter 10 42 Glencoe Algebra 2 Chapter 10 43 Glencoe Algebra 2
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-6 Study Guide and Intervention (continued) 10-6 Skills Practice
Conic Sections Conic Sections
Identify Conic Sections If you are given an equation of the form Write each equation in standard form. State whether the graph of the equation is
a parabola, circle, ellipse, or hyperbola. Then graph the equation.
Ax2 ϩ Bxy ϩ Cy2 ϩ Dx ϩ Ey ϩ F ϭ 0, with B ϭ 0, Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
you can determine the type of conic section just by considering the values of A and C. Refer 1. x2 Ϫ 25y2 ϭ 25 hyperbola 2. 9x2 ϩ 4y2 ϭ 36 ellipse 3. x2 ϩ y2 Ϫ 16 ϭ 0 circle
to the following chart. Lesson 10-6ᎏ2x52 Ϫ ᎏy12 ϭ 1ᎏx42 ϩ ᎏy92 ϭ 1x2 ϩ y2 ϭ 16
Relationship of A and C Type of Conic Section yy y
A ϭ 0 or C ϭ 0, but not both. parabola 4
AϭC circle
A and C have the same sign, but A C. ellipse 2
A and C have opposite signs. hyperbola
–8 –4 O 4 8x Ox Ox
–2
Example Without writing the equation in standard form, state whether the
–4
graph of each equation is a parabola, circle, ellipse, or hyperbola. Answers (Lesson 10-6)
a. 3x2 Ϫ 3y2 ϩ 5x ϩ 12 ϭ 0 b. y2 ϭ 7y Ϫ 2x ϩ 13 4. x2 ϩ 8x ϩ y2 ϭ 9 circle 5. x2 ϩ 2x Ϫ 15 ϭ y parabola 6. 100x2 ϩ 25y2 ϭ 400
(x ϩ 4)2 ϩ y2 ϭ 25 ᎏx42 y2
A ϭ 3 and C ϭ Ϫ3 have opposite signs, so A ϭ 0, so the graph of the equation is y ϭ (x ϩ 1)2 Ϫ 16 ϩ ᎏ16 ϭ 1 ellipse
the graph of the equation is a hyperbola. a parabola. y
8 yy
4 –8 –4 O 4 8 x
Exercises –4
A21 Without writing the equation in standard form, state whether the graph of each –8 –4 O 4 8x –8 O x
equation is a parabola, circle, ellipse, or hyperbola. –4 –12
1. x2 ϭ 17x Ϫ 5y ϩ 8 2. 2x2 ϩ 2y2 Ϫ 3x ϩ 4y ϭ 5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. –8 –16
parabola circle Without writing the equation in standard form, state whether the graph of each
equation is a parabola, circle, ellipse, or hyperbola.
3. 4x2 Ϫ 8x ϭ 4y2 Ϫ 6y ϩ 10 4. 8(x Ϫ x2) ϭ 4(2y2 Ϫ y) Ϫ 100
7. 9x2 ϩ 4y2 ϭ 36 ellipse 8. x2 ϩ y2 ϭ 25 circle
hyperbola circle
9. y ϭ x2 ϩ 2x parabola 10. y ϭ 2x2 Ϫ 4x Ϫ 4 parabola
5. 6y2 Ϫ 18 ϭ 24 Ϫ 4x2 6. y ϭ 27x Ϫ y2
11. 4y2 Ϫ 25x2 ϭ 100 hyperbola 12. 16x2 ϩ y2 ϭ 16 ellipse
ellipse parabola
7. x2 ϭ 4( y Ϫ y2) ϩ 2x Ϫ 1 8. 10x Ϫ x2 Ϫ 2y2 ϭ 5y
ellipse ellipse
9. x ϭ y2 Ϫ 5y ϩ x2 Ϫ 5 10. 11x2 Ϫ 7y2 ϭ 77
circle hyperbola
11. 3x2 ϩ 4y2 ϭ 50 ϩ y2 12. y2 ϭ 8x Ϫ 11 13. 16x2 Ϫ 4y2 ϭ 64 hyperbola 14. 5x2 ϩ 5y2 ϭ 25 circle
circle parabola 15. 25y2 ϩ 9x2 ϭ 225 ellipse 16. 36y2 Ϫ 4x2 ϭ 144 hyperbola
13. 9y2 Ϫ 99y ϭ 3(3x Ϫ 3x2) 14. 6x2 Ϫ 4 ϭ 5y2 Ϫ 3 17. y ϭ 4x2 Ϫ 36x Ϫ 144 parabola 18. x2 ϩ y2 Ϫ 144 ϭ 0 circle
circle hyperbola
Glencoe Algebra 2 15. 111 ϭ 11x2 ϩ 10y2 16. 120x2 Ϫ 119y2 ϩ 118x Ϫ 117y ϭ 0 19. (x ϩ 3)2 ϩ ( y Ϫ 1)2 ϭ 4 circle 20. 25y2 Ϫ 50y ϩ 4x2 ϭ 75 ellipse
ellipse hyperbola 21. x2 Ϫ 6y2 ϩ 9 ϭ 0 hyperbola 22. x ϭ y2 ϩ 5y Ϫ 6 parabola
17. 3x2 ϭ 4y2 ϩ 12 18. 150 Ϫ x2 ϭ 120 Ϫ y 23. (x ϩ 5)2 ϩ y2 ϭ 10 circle 24. 25x2 ϩ 10y2 Ϫ 250 ϭ 0 ellipse
hyperbola parabola
Chapter 10 44 Glencoe Algebra 2 Chapter 10 45 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-6 Practice 10-6 Word Problem Practice
Conic Sections Conic Sections 4. MIRROR A painter used a can of spray
paint to make an image. The boundary
Write each equation in standard form. State whether the graph of the equation is 1. MISSING INFORMATION Rick began of the image is described by the equation
a parabola, circle, ellipse, or hyperbola. Then graph the equation. reading a book on conic sections. He
came to this passage and discovered an 4x2 Ϫ 16x ϩ y2 Ϫ 6y ϩ 21 ϭ 0.
inkblot covering part of an equation.
Put this equation into standard form
For example, although it may not be obvious, and describe whether the curve is a
the equation below describes a circle. circle, ellipse, parabola, or hyperbola.
7x2Ϫ12xϩ y2Ϫ16yϪ94ϭ0 (x ؊ 2)2 ؉ ᎏ(y ؊4 3)2 ؍1, ellipse
To see that it is a circle, observe that the NONSTANDARD EQUATIONS
Based on the information in the passage For Exercises 5–7, use the following
and your own knowledge of conic information.
sections, what number is being covered
by the inkblot? Consider the equation xy ϭ 1.
7
1. y2 ϭ Ϫ3x 2. x2 ϩ y2 ϩ 6x ϭ 7 3. 5x2 Ϫ 6y2 Ϫ 30x Ϫ 12y ϭ Ϫ9 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
parabola circle hyperbola Lesson 10-6
x ϭ Ϫᎏ13 y 2 (x ϩ 3)2 ϩ y2 ϭ 16 ᎏ(x Ϫ6 3)2 Ϫ ᎏ(y ϩ5 1)2 ϭ 1
y y y
O x
Ox Ox
A22 4. 196y2 ϭ 1225 Ϫ 100x2 5. 3x2 ϭ 9 Ϫ 3y2 Ϫ 6y 6. 9x2 ϩ y2 ϩ 54x Ϫ 6y ϭ Ϫ81 2. HEADLIGHTS The light from the 5. Are there any solutions of this equation Answers (Lesson 10-6)
headlight of a car is in the shape of that lie on the x- or y-axis?
ellipse circle ellipse a cone. The axis of the cone is parallel
ᎏ12x.225 ϩ ᎏ6y.225 ϭ 1 x2 ϩ (y ϩ 1)2 ϭ 4 ᎏ(x ϩ1 3)2 ϩ ᎏ(y Ϫ9 3)2 ϭ 1 to the ground. What shape does the no
edge of the lit region form on the road,
y y y assuming that the road is flat and level? 6. Sketch a graph of the solutions of the
hyperbola equation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
Ox Ox
Ox
Without writing the equation in standard form, state whether the graph of each 3. REASONING Jason has been struggling Ox
equation is a parabola, circle, ellipse, or hyperbola. with conic sections. He decides he needs
more practice, but he needs to have a 7. Assuming that the equation represents a
7. 6x2 ϩ 6y2 ϭ 36 8. 4x2 Ϫ y2 ϭ 16 9. 9x2 ϩ 16y2 Ϫ 64y Ϫ 80 ϭ 0 way of making practice equations. He conic section, based on the graph, which
decides to use an equation of the form type of conic section is it?
circle hyperbola ellipse
Ax2 ϩ By2 ϭ 1, a hyperbola
10. 5x2 ϩ 5y2 Ϫ 45 ϭ 0 11. x2 ϩ 2x ϭ y 12. 4y2 Ϫ 36x2 ϩ 4x Ϫ 144 ϭ 0
where A and B are determined by rolling
circle parabola hyperbola a pair of dice. After several rolls, he
begins to realize that this system is not
Glencoe Algebra 2 13. ASTRONOMY A satellite travels in an hyperbolic orbit. It reaches the vertex of its orbit good enough because some conic sections
at (5, 0) and then travels along a path that gets closer and closer to the line y ϭ ᎏ25 x. never appear. Which types of conic
Write an equation that describes the path of the satellite if the center of its hyperbolic section cannot occur using his method?
orbit is at (0, 0).
ᎏ2x52 Ϫ ᎏy42 ϭ 1 parabolas and hyperbolas are not
possible
Chapter 10 46 Glencoe Algebra 2 Chapter 10 47 Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-6 Enrichment 10-7 Lesson Reading Guide
Parabolic Football Solving Quadratic Systems
A parabola is defined as all the points (x, y) in the plane whose distance from a fixed point, Get Ready for the Lesson
called the focus is the same as its distance from a fixed line, called the directrix. Examples
of parabolas are the cables on a suspension bridge, satellite dishes, and the flight path of a Read the introduction to Lesson 10-7 in your textbook.
football during a kick-off.
The figure in your textbook shows that the spaceship hits the circular force field in two
10 20 30 40 50 40 30 20 10 points. Is it possible for the spaceship to hit the force field in either fewer or more than
two points? State all possibilities and explain how these could happen. Sample answer:
The spaceship could hit the force field in zero points if the spaceship
missed the force field all together. The spaceship could also hit the force
field in one point if the spaceship just touched the edge of the force field.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Read the Lesson Answers (Lessons 10-6 and 10-7)
Lesson 10-7
10 20 30 40 50 40 30 20 10 1. Draw a sketch to illustrate each of the following possibilities.
At the kick-off at the beginning of a football game the ball is placed on the 40-yard line a. a parabola and a line b. an ellipse and a circle c. a hyperbola and a
of the kicking team. Suppose the receiving team catches the ball on the goal line. Assume that intersect in that intersect in line that intersect in
the 50-yard line has coordinates (0, 0) and the 40-yard line of the kicking team has 2 points 4 points 1 point
coordinates (Ϫ10, 0).
y y y
Ox Ox Ox
A23 1. Determine which equation of the parabola best describes this situation. For the other Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
choices explain why they do not make sense to the situation.
a. y ϭ 50x(x ϩ 10) Incorrect. This says the goal line is the 2. Consider the following system of equations.
50-yard line x2 ϭ 3 ϩ y2
2x2 ϩ 3y2 ϭ 11
b. y ϭ 100(x Ϫ 50)(x ϩ 10) Incorrect. Wrong direction and way too high.
c. y ϭ Ϫ ᎏ14 (x Ϫ 50)(x ϩ 10) Correct a. What kind of conic section is the graph of the first equation? hyperbola
d. y ϭ Ϫ20(x Ϫ 50)(x Ϫ 60) Incorrect. Too high, not right yard lines.
b. What kind of conic section is the graph of the second equation? ellipse
2. During the same game, the quarterback throws a forward pass from the 50-yard line to
his receiver on the 25-yard line. Assuming the ball follows the path of a parabola, write c. Based on your answers to parts a and b, what are the possible numbers of solutions
an equation modeling the flight path of the ball from quarterback to receiver.
that this system could have? 0, 1, 2, 3, or 4
Answer may vary based on orientation.
Remember What You Learned
Glencoe Algebra 2 3. The team did not pick up the first down, so they elect to try a field goal. Fortunately,
3. Suppose that the graph of a quadratic inequality is a region whose boundary is a circle.
one of the assistant coaches is a part-time mathematician and found an equation that How can you remember whether to shade the interior or the exterior of the circle?
describes their kicker as: y ϭ Ϫ0.2x2 ϩ 8x. The line of scrimmage is the 25-yard line, so
the ball will be placed on the 32-yard line for the kick. Add 10 more yards for the depth Sample answer: The solutions of an inequality of the form x2 ϩ y 2 Ͻ r2
are all points that are less than r units from the origin, so the graph is
of the end zone (goal line to the goal post), making it a 42-yard field goal attempt. Will the interior of the circle. The solutions of an inequality of the form
x2 ϩ y 2 Ͼ r 2 are the points that are more than r units from the origin, so
your kicker make it? the graph is the exterior of the circle.
Setting the vertex at the origin, then the focus is located at 0, ᎏ18 .
Chapter 10 48 Glencoe Algebra 2 Chapter 10 49 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-7 Study Guide and Intervention 10-7 Study Guide and Intervention (continued)
Solving Quadratic Systems Solving Quadratic Systems
Systems of Quadratic Equations Like systems of linear equations, systems of Systems of Quadratic Inequalities Systems of quadratic inequalities can be solved
quadratic equations can be solved by substitution and elimination. If the graphs are a conic by graphing.
section and a line, the system will have 0, 1, or 2 solutions. If the graphs are two conic
sections, the system will have 0, 1, 2, 3, or 4 solutions. Example 1 Solve the system of inequalities by graphing.
Example Solve the system of equations. y ϭ x2 Ϫ 2x Ϫ 15 x2 ϩ y2 Յ 25 y
x ϩ y ϭ Ϫ3 Ox
xϪ 5 2 y2 Ն 25
ᎏ2 ᎏ4
ϩ
Rewrite the second equation as y ϭ Ϫx Ϫ 3 and substitute into the first equation. The graph of x2 ϩ y2 Յ 25 consists of all points on or inside
the circle with center (0, 0) and radius 5.The graph of
x Ϫ ᎏ52 2 ϩ y2 Ն ᎏ245 consists of all points on or outside the
Ϫx Ϫ 3 ϭ x2 Ϫ 2x Ϫ 15 Add x ϩ 3 to each side. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0 ϭ x2 Ϫ x Ϫ 12 Factor. circle with center ᎏ52 , 0 and radius ᎏ25 . The
Lesson 10-7
0 ϭ (x Ϫ 4)(x ϩ 3)
Use the Zero Product property to get solution of the Answers (Lesson 10-7)
x ϭ 4 or x ϭ Ϫ3.
system is the set of points in both regions.
Substitute these values for x in x ϩ y ϭ Ϫ3:
4 ϩ y ϭ Ϫ3 or Ϫ3 ϩ y ϭ Ϫ3 Example 2 Solve the system of inequalities by graphing. y
Ox
y ϭ Ϫ7 yϭ0 x2 ϩ y2 Յ 25
The solutions are (4, Ϫ7) and (Ϫ3, 0). y2 Ϫ x2 Ͼ 1
ᎏ4 ᎏ9
A24 Exercises The graph of x2 ϩ y2 Յ 25 consists of all points on or inside
the circle with center (0, 0) and radius 5.The graph of
Find the exact solution(s) of each system of equations. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
ᎏy42 Ϫ ᎏx92 Ͼ 1 are the points “inside” but not on the branches of
1. yϭ x2 Ϫ 5 2. x2 ϩ ( y Ϫ 5)2 ϭ 25 the hyperbola shown. The solution of the system is the set of
yϭ x Ϫ 3 y ϭ Ϫx2 points in both regions.
(2, Ϫ1), (Ϫ1, Ϫ4) (0, 0) Exercises
Solve each system of inequalities below by graphing.
3. x2 ϩ ( y Ϫ 5)2 ϭ 25 4. x2 ϩ y2 ϭ 9 1. ᎏ1x62 ϩ ᎏy42 Յ 1 2. x2 ϩ y2 Յ 169 3. y Ն (x Ϫ 2)2
y ϭ x2 x2 ϩ y ϭ 3 y Ͼ ᎏ12 x Ϫ 2 x2 ϩ 9y2 Ն 225 (x ϩ 1)2 ϩ ( y ϩ 1)2 Յ 16
(0, 0), (3, 9), (Ϫ3, 9) (0, 3), (͙5ෆ, Ϫ2), (Ϫ͙5ෆ, Ϫ2) y
yy Ox
5. x2 Ϫ y2 ϭ 1 6. y ϭ x Ϫ 3 12
x2 ϩ y2 ϭ 16 x ϭ y2 Ϫ 4
6
ᎏ͙23ෆ4 , ᎏ͙23ෆ0 , ᎏ͙23ෆ4 , Ϫᎏ͙2ෆ30 , ᎏ7 ϩ2͙ᎏෆ29 , ᎏ1 ϩ2͙ᎏෆ29 ,
Glencoe Algebra 2 O x –12 –6 O 6 12 x
Ϫᎏ͙3ෆ4 , ᎏ͙3ෆ0 , Ϫᎏ͙3ෆ4 , Ϫᎏ͙ෆ30 ᎏ7 Ϫ2͙ᎏ2ෆ9 , ᎏ1 Ϫ2͙ᎏ2ෆ9 –6
22 22
–12
Chapter 10 50 Glencoe Algebra 2 Chapter 10 51 Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-7 Skills Practice 10-7 Practice
Solving Quadratic Systems Solving Quadratic Systems
Find the exact solution(s) of each system of equations. Find the exact solution(s) of each system of equations.
ᎏ91ᎏ, ᎏ13ᎏ
1. y ϭ x Ϫ 2 (0, Ϫ2), (1, Ϫ1) 2. y ϭ x ϩ 3 (Ϫ1, 2), 3. y ϭ 3x (0, 0), 1. (x Ϫ 2)2 ϩ y2 ϭ 5 2. x ϭ 2( y ϩ 1)2 Ϫ 6 3. y2 Ϫ 3x2 ϭ 6 4. x2 ϩ 2y2 ϭ 1
x ϭ y2 y ϭ Ϫx ϩ 1
y ϭ x2 Ϫ 2 y ϭ 2x2 (1.5, 4.5) xϪyϭ1 xϩyϭ3 y ϭ 2x Ϫ 1
(1, 0), ᎏ13 , ᎏ23
(0, Ϫ1), (3, 2) (2, 1), (6.5, Ϫ3.5) (Ϫ1, Ϫ3), (5, 9)
8. y2 ϭ 10 Ϫ 6x2
4. y ϭ x (͙2ෆ, ͙ෆ2), 5. x ϭ Ϫ5 (Ϫ5, 0) 6. y ϭ 7 no solution 5. 4y2 Ϫ 9x2 ϭ 36 6. y ϭ x2 Ϫ 3 7. x2 ϩ y2 ϭ 25 4y2 ϭ 40 Ϫ 2x2
4x2 Ϫ 9y2 ϭ 36 x2 ϩ y2 ϭ 9 4y ϭ 3x
x2 ϩ y2 ϭ 4 (Ϫ͙2ෆ, Ϫ͙2ෆ) x2 ϩ y2 ϭ 25 x2 ϩ y2 ϭ 9 (0, Ϯ͙ෆ10 )
no solution
7. y ϭ Ϫ2x ϩ 2 (2, Ϫ2), 8. x Ϫ y ϩ 1 ϭ 0 (1, 2) 9. y ϭ 2 Ϫ x (0, 2), (3, Ϫ1) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.(0, Ϫ3), (Ϯ͙5ෆ, 2) (4, 3), (Ϫ4, Ϫ3)
y2 ϭ 2x ᎏ12 , 1 y2 ϭ 4x y ϭ x2 Ϫ 4x ϩ 2 9. x2 ϩ y2 ϭ 25 Lesson 10-710. 4x2 ϩ 9y2 ϭ 3611.xϭϪ( y Ϫ 3)2 ϩ 212.ᎏx92 Ϫ ᎏ1y62 ϭ 1
x ϭ 3y Ϫ 5 2x2 Ϫ 9y2 ϭ 18 x ϭ ( y Ϫ 3)2 ϩ 3 x2 ϩ y2 ϭ 9
11. y ϭ 3x2 (0, 0) 12. y ϭ x2 ϩ 1 (Ϫ1, 2),
10. y ϭ x Ϫ 1 no solution y ϭ Ϫx2 ϩ 3 (1, 2) (Ϫ5, 0), (4, 3) (Ϯ3, 0) no solution (Ϯ3, 0) Answers (Lesson 10-7)
y ϭ Ϫ3x2
y ϭ x2
13. y ϭ 4x (Ϫ1, Ϫ4), (1, 4) 14. y ϭ Ϫ1 (0, Ϫ1) 15. 4x2 ϩ 9y2 ϭ 36 (Ϫ3, 0), 13. 25x2 ϩ 4y2 ϭ 100 14. x2 ϩ y2 ϭ 4 15. x2 Ϫ y2 ϭ 3
x2 Ϫ 9y2 ϭ 9 (3, 0) x ϭ Ϫᎏ25 ᎏx42 ϩ ᎏy82 ϭ 1 y2 Ϫ x2 ϭ 3
4x2 ϩ y2 ϭ 20 4x2 ϩ y2 ϭ 1
no solution (Ϯ2, 0) no solution
A25 16. 3( y ϩ 2)2 Ϫ 4(x Ϫ 3)2 ϭ 12 17. x2 Ϫ 4y2 ϭ 4 (Ϫ2, 0), 18. y2 Ϫ 4x2 ϭ 4 no x2 y2 17. x ϩ 2y ϭ 3 18. x2 ϩ y2 ϭ 64
y ϭ Ϫ2x ϩ 2 (0, 2), (3, Ϫ4) x2 ϩ y2 ϭ 4 (2, 0) ᎏ7 ᎏ7 x2 ϩ y2 ϭ 9 x2 Ϫ y2 ϭ 8
y ϭ 2x solution 16. ϩ ϭ 1
(3, 0), Ϫᎏ59 , ᎏ152 (Ϯ6, Ϯ2͙ෆ7)
3x2 Ϫ y2 ϭ 9
Solve each system of inequalities by graphing. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. (Ϯ2, Ϯ͙ෆ3)
19. y Յ 3x Ϫ 2 20. y Յ x 21. 4y2 ϩ 9x2 Ͻ 144 Solve each system of inequalities by graphing.
x2 ϩ y2 Ͻ 16 y Ն Ϫ2x2 ϩ 4 x2 ϩ 8y2 Ͻ 16
19. y Ն x2 20. x2 ϩ y2 Ͻ 36 21. ᎏ( y Ϫ163)2 ϩ ᎏ(x ϩ4 2)2 Յ 1
y y y y Ͼ Ϫx ϩ 2 x2 ϩ y2 Ն 16 (x ϩ 1)2 ϩ ( y Ϫ 2)2 Յ 4
8
yy y
4 8
Ox Ox –8 –4 O 4 8x 4
–4
–8 –4 O 4 8x
–8 –4
Ox –8 Ox
22. GARDENING An elliptical garden bed has a path from point A to y 22. GEOMETRY The top of an iron gate is shaped like half an A B
point B. If the bed can be modeled by the equation x2 ϩ 3y2 ϭ 12 ellipse with two congruent segments from the center of the
and the path can be modeled by the line y ϭ Ϫᎏ31 x, what are the A ellipse to the ellipse as shown. Assume that the center of (0, 0)
the ellipse is at (0, 0). If the ellipse can be modeled by the
Glencoe Algebra 2 coordinates of points A and B? (Ϫ3, 1) and (3, Ϫ1) Ox
equation x2 ϩ 4y2 ϭ 4 for y Ն 0 and the two congruent
B segments can be modeled by y ϭ ᎏ͙2ෆ3 x and y ϭ Ϫᎏ͙2ෆ3 x, Ϫ1, ᎏ͙23ෆ and 1, ᎏ͙2ෆ3
what are the coordinates of points A and B?
Chapter 10 52 Glencoe Algebra 2 Chapter 10 53 Glencoe Algebra 2
Answers
Chapter 10 NAME ______________________________________________ DATE______________ PERIOD _____ NAME ______________________________________________ DATE______________ PERIOD _____
10-7 Word Problem Practice 10-7 Enrichment
Solving Quadratic Systems
1. GRAPHIC DESIGN A graphic designer 4. COLLISION AVOIDANCE An object Graphing Quadratic Equations with xy-Terms
is drawing an ellipse and a line. The is traveling along a hyperbola given by
ellipse is drawn so that it appears on top the equation ᎏx92 Ϫ ᎏ3y62 ϭ 1. A probe is You can use a graphing calculator to examine graphs of quadratic equations
of the line. In order to determine if the that contain xy-terms.
line is partially covered by the ellipse, launched from the origin along a
the program solves for simultaneous straight-line path. Mission planners Example Use a graphing calculator to display y
solutions of the equations of the line and want the probe to get closer and closer 2
the ellipse. Complete the following table. to the object, but never hit it. There are the graph of x2 ϩ xy ϩ y2 ϭ 4. 1
two straight lines that meet their
No. of Intersections Covered? Y/N criteria. What are they? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Solve the equation for y in terms of x by using the
0 quadratic formula.
1 N y ϭ 2x and y ϭ Ϫ2x Lesson 10-7
2 N or Y ok y2 ϩ xy ϩ (x2 Ϫ 4) ϭ 0
–2 –1 O 1 2x
Y –1
To use the formula, let a ϭ 1, b ϭ x, and c ϭ (x2 Ϫ 4). Answers (Lesson 10-7)
y ϭ ᎏϪx Ϯ ͙ෆxᎏ2 Ϫ2 4(ෆ1)ᎏ(x2 Ϫෆ4) –2
y ϭ ᎏϪx Ϯᎏ͙2ෆ16 Ϫᎏ3ෆx2
2. ORBITS An asteroid travels in an TANGENTS For Exercises 5 and 6, use
A26 elliptical orbit. If the orbit of Earth is the following information. To graph the equation on the graphing calculator, enter the two equations:
also an ellipse, what is the maximum y ϭ ᎏϪx ϩᎏ͙2ෆ16 Ϫᎏෆ3x2 and y ϭ ᎏϪx Ϫᎏ͙21ෆ6 Ϫᎏෆ3x2
number of times the asteroid could cross An architect wants a straight path to run
the orbit of Earth? from the origin of a coordinate plane to the Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Exercises
4 edge of an elliptically shaped patio so that
the pathway forms a tangent to the ellipse. Use a graphing calculator to graph each equation. State the type
3. CIRCLES An artist is commissioned to The ellipse is given by the equation of curve each graph represents.
complete a painting of only circles. She
wants to include all possible ways circles ᎏ(x Ϫ126)2 ϩ ᎏ9y62 ϭ 1. 1. y2 ϩ xy ϭ 8 2. x2 ϩ y2 Ϫ 2xy Ϫ x ϭ 0
can relate. What are the possible
numbers of intersection points between 5. Using the equation y ϭ mx to describe hyperbola parabola
two circles? For each case, sketch two the path, substitute into the equation
distinct circles that intersect with the for the ellipse to get a quadratic
corresponding number of points. Explain equation in x.
why more intersections are not possible.
1 ؉ ᎏm82 x2 ؊ 12x ؉ 24 ؍0
0 or 3. x2 Ϫ xy ϩ y2 ϭ 15 4. x2 ϩ xy ϩ y2 ϭ Ϫ9
1 or
2 ellipse graph is л
6. Solve for m in the equation you found
for Exercise 5.
m ؍2 or ؊2
Glencoe Algebra 2 5. 2x2 Ϫ 2xy Ϫ y2 ϩ 4x ϭ 20 6. x2 Ϫ xy Ϫ 2y2 ϩ 2x ϩ 5y Ϫ 3 ϭ 0
hyperbola two intersecting lines
A circle is determined by 3
points. Therefore, any two circles
that share 3 or more points must
be identical and not distinct.
Chapter 10 54 Glencoe Algebra 2 Chapter 10 55 Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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