C H A P T ER Hyperbolas Flashlights concentrate beams of light by bouncing the rays from a light source off a reflector. The cross-section of a reflector can be described as hyperbola with the light source at its focus. You will use the mathematics of hyperbolas to describe the design of a flashlight..1 These Hyperbolas Like Being the Center of Attention Hyperbolas Centered at the Origin p. 547.2 And Then a Step to the Right! Hyperbolas Not Centered at the Origin p. 569.3 Graphs, Equations, and Key Characteristics of Hyperbolas Forms of Hyperbolas p. 583.4 Floodlights and Flashlights and LORAN Hyperbolas and Problem Solving p. 595 Chapter l Hyperbolas 545
546 Chapter l Hyperbolas
.1 These Hyperbolas Like Being the Center of Attention Hyperbolas Centered at the Origin Objectives In this lesson you will: l Graph a hyperbola using its definition and intersecting circles. l Graph a hyperbola centered at the origin from its equation in standard form. l Write the general and standard form of the equation of a hyperbola centered at the origin. l Determine the key characteristics of hyperbolas: transverse axes, conjugate axes, major and minor axes, semi-major and semi-minor axes, vertices and co-vertices, foci, eccentricity, and asymptotes of a hyperbola from its graph or equation. Key Terms l hyperbola l foci / focus l general form of a hyperbola l standard form of a hyperbola l major axis l transverse axis l vertices l semi-major axis l minor axis l conjugate axis l co-vertices l semi-minor axis l asymptote l eccentricity Problem 1 Hyperbolas as Sets of Points A hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points is constant. The foci of a hyperbola are the two fixed points. The singular form of foci is focus. 1. How is the definition of a hyperbola similar to the definition of an ellipse? How is it different? Lesson.1 l Hyperbolas Centered at the Origin 547
2. The centers for two sets of circles are represented by point X and point Y. The radius of the smallest circle with center X is 1 unit with the radius of each successive circle increasing by 1 unit. Likewise, the radius of the smallest circle with center Y is 1 unit with the radius of each successive circle increasing by 1 unit. X Y a. What is the distance from the star to point X? How do you know? b. What is the distance from the star to point Y? How do you know? c. What is the difference of the distances from the two given points, X and Y, to the star? d. Graph 17 additional points such that the difference of the distances from the point to point X and point Y is equal to the distance in Question 2, part (c). e. Draw a hyperbola by connecting the points with a smooth curve. 548 Chapter l Hyperbolas
Problem 2 Equations of Hyperbolas 1. A hyperbola is defined such that the difference of the distances from (5, 0) and ( 5, 0) to any point on the hyperbola is 6 units. One point on the hyperbola is labeled as (x, y). Determine the equation of the hyperbola by completing the following steps. y 8 6 d 2 d 1 = 6 (x, y) 4 d 1 d 2 2 ( 5, 0) (5, 0) 8 6 4 2 2 4 6 8 2 x 4 6 8 a. Let d 1 represent the distance from (x, y) to ( 5, 0). Write an equation using the distance formula to represent d 1. Simplify the equation. b. Let d 2 represent the distance from (x, y) to (5, 0). Write an equation using the distance formula to represent d 2. Simplify the equation. c. What do you know about the difference of d 1 and d 2? Lesson.1 l Hyperbolas Centered at the Origin 549
d. Write an equation for the hyperbola using Question 1, parts (a) through (c). Simplify the equation so that one side of the equation is equal to 1. The general form of a hyperbola centered at the origin is an equation of the form Ax 2 By 2 E 0. The standard form of a hyperbola centered at the origin is an equation of the form x2 a y2 2 b 1 or y2 2 a x2 1. 2 2 b 2. Write the equation of the hyperbola from Question 1 in general form and standard form. 3. What are the x-intercepts of the hyperbola? 550 Chapter l Hyperbolas
4. What are the y-intercepts of the hyperbola? 5. How many points on the hyperbola have an x-coordinate of 4? Calculate the coordinates of each point. 6. How many points on the hyperbola have an x-coordinate of 4? Calculate the coordinates of each point. 7. How many points on the hyperbola have an x-coordinate of 8? Calculate the coordinates of each point. Lesson.1 l Hyperbolas Centered at the Origin 551
8. How many points on the hyperbola have an x-coordinate of 8? Calculate the coordinates of each point. 9. Sketch the hyperbola on the grid shown using the points from Questions 3 through 8. y 8 6 4 2 8 6 4 2 2 4 6 8 2 x 4 6 8 10. Describe the symmetry of the hyperbola. Problem 3 Key Characteristics of a Hyperbola Many of the terms used to describe ellipses are also used to describe hyperbolas. However, hyperbolas are open figures and ellipses are closed figures, so the definition of these terms is slightly different for hyperbolas. The major axis of a hyperbola is the shortest segment that connects two points on the hyperbola and passes through the center of the hyperbola. The major axis is also referred to as the transverse axis. The vertices of a hyperbola are the endpoints of the major axis. Both vertices lie on the hyperbola. The semi-major axis is a segment that connects the center of a hyperbola to one of the vertices. 552 Chapter l Hyperbolas
The minor axis of a hyperbola passes through its center and is perpendicular to the major axis. The minor axis is also referred to as the conjugate axis. The co-vertices of a hyperbola are the endpoints of the minor axis. Neither co-vertex lies on the hyperbola but are useful when graphing a parabola. The semi-minor axis is a segment that connects the center of a hyperbola to one of the co-vertices. Unlike an ellipse, the major axis of a hyperbola may be shorter than the minor axis. When the major axis is horizontal, the equation of a hyperbola centered at the origin is x2 y2 1. When the major axis is vertical, the equation of a hyperbola centered 2 2 a b at the origin is y2 a x2 1. In both equations, a represents the length of the 2 2 b semi-major axis and b represents the length of the semi-minor axis. 1. Consider the hyperbola represented by the equation x2 y2 1. 4 25 a. Is the major axis along the x-axis or the y-axis? How can you tell? b. What are the coordinates of the vertices? Plot the vertices on the grid. y 8 6 4 2 8 6 4 2 2 4 6 8 2 4 6 8 c. What are the coordinates of the co-vertices? Plot the co-vertices on the grid in part (b). x Lesson.1 l Hyperbolas Centered at the Origin 553
Take Note An asymptote is a line that is approached by the graph of a function. The graph does not touch or cross the line at any point, and the distance between the graph and the line approaches zero. d. Draw dashed vertical segments extending through each vertex and dashed horizontal segments extending through each co-vertex to form a rectangle. Draw a dashed line passing through each pair of opposite angles of the rectangle. These lines are the asymptotes of the hyperbola. The ends of the hyperbola will get very close to these lines but never intersect e. Calculate an equation for each asymptote line. f. How do the slopes of the asymptotes compare to the lengths of the semi-major and semi-minor axes? g. Draw the hyperbola using the vertices and the asymptotes as a guide. 2. Consider the hyperbola represented by the equation y2 x2 1. 9 49 a. Is the major axis along the x-axis or the y-axis? How can you tell? 554 Chapter l Hyperbolas
b. What are the coordinates of the vertices? Plot the vertices on the grid. y 8 6 4 2 8 6 4 2 2 4 6 8 2 x 4 6 8 c. What are the coordinates of the co-vertices? Plot the co-vertices on the grid in part (b). d. Draw dashed vertical segments extending through each vertex and dashed horizontal segments extending through each co-vertex to form a rectangle. Draw a dashed line passing through each pair of opposite angles of the rectangle. These lines are the asymptotes of the hyperbola. The ends of the hyperbola will get very close to these lines but never intersect. Lesson.1 l Hyperbolas Centered at the Origin 555
e. Calculate an equation for each asymptote line. f. How does the slope of the asymptotes compare to the lengths of the semi-major and semi-minor axes? g. Draw the hyperbola using the vertices and the asymptotes as a guide. 3. Let a represent the length of the semi-major axis. Let b represent the length of the semi-minor axis. a. What is the standard form of a hyperbola with the major axis along the x-axis? b. What is the standard form of a hyperbola with the major axis along the y-axis? c. What are the coordinates of the vertices for a hyperbola with the major axis along the x-axis? 556 Chapter l Hyperbolas
d. What are the coordinates of the vertices for a hyperbola with the major axis along the y-axis? e. What are the coordinates of the co-vertices for a hyperbola with the major axis along the x-axis? f. What are the coordinates of the co-vertices for a hyperbola with the major axis along the y-axis? g. What are the intercepts of a hyperbola centered at the origin with its major axis along the x-axis? h. What are the intercepts of a hyperbola centered at the origin with its major axis along the y-axis? i. What are the equations for the asymptotes of a hyperbola centered at the origin with its major axis along the x-axis? Lesson.1 l Hyperbolas Centered at the Origin 557
j. What are the equations for the asymptotes of a hyperbola centered at the origin with its major axis along the y-axis? 4. Consider the hyperbola represented by the equation x2 y2 1. 64 100 a. Is the major axis along the x-axis or the y-axis? b. Is the minor axis along the x-axis or the y-axis? c. What is the length of the semi-major axis and the major axis? d. What is the length of the semi-minor axis and the minor axis? e. What are the coordinates of the vertices and the co-vertices? f. What are the intercepts? g. What are the equations of the asymptotes? 558 Chapter l Hyperbolas
h. Sketch the hyperbola. Label the intercepts and asymptotes. Problem 4 Foci, Vertices, and Co-Vertices (Oh My) The foci of a hyperbola always lie on the major axis of the hyperbola. There is a numeric relationship between the foci, vertices, and co-vertices. 1. Let a represent the length of the semi-major axis. Let b represent the length of the semi-minor axis. Let c represent the distance from the center to a focus. Label the coordinates of the foci, vertices and co-vertices of the hyperbola. Lesson.1 l Hyperbolas Centered at the Origin 559
2. Describe what was done algebraically for each step in determining the numeric relationship between a, b, and c. Statements Reasons a. (x c) 2 y 2 (x c) 2 y 2 2a b. (x c) 2 y 2 2a (x c) 2 y 2 c. ( (x c) 2 y 2 ) 2 ( 2a (x c) 2 y 2 ) 2 d. (x c) 2 y 2 4a 2 4a (x c) 2 y 2 (x c) 2 y 2 e. x 2 2cx c 2 y 2 4a 2 4a (x c) 2 y 2 x 2 2cx c 2 y 2 f. 4cx 4a 2 4a (x c) 2 y 2 g. cx a 2 a (x c) 2 y 2 h. (cx a 2 ) 2 ( a (x c) 2 y 2 ) 2 i. c 2 x 2 2a 2 cx a 4 a 2 (x 2 2cx c 2 y 2 ) j. c 2 x 2 2a 2 cx a 4 a 2 x 2 2a 2 cx a 2 c 2 a 2 y 2 k. c 2 x 2 a 2 x 2 a 2 y 2 a 2 c 2 a 4 l. (c 2 a 2 )x 2 a 2 y 2 a 2 (c 2 a 2 ) m. x2 a y 2 2 c 2 a 1 2 560 Chapter l Hyperbolas
3. Write an equation describing the relationship between a, b, and c. 4. How is the relationship between a, b, and c for a hyperbola different than the relationship for an ellipse? 5. Determine the location of the foci for each hyperbola. a. x2 25 y2 100 1 b. y2 64 x2 9 1 c. y2 10 x2 20 1 Lesson.1 l Hyperbolas Centered at the Origin 561
Problem 5 Eccentricity The eccentricity, e, of a hyperbola is a measure of the angle of intersection of its asymptotes and, therefore, also a measure of the shape of a hyperbola. It is calculated as c c a, such that a 1. 1. Calculate the eccentricity of each hyperbola. Then sketch each hyperbola. a. x2 25 y2 100 1 562 Chapter l Hyperbolas
b. y2 64 x2 9 1 Lesson.1 l Hyperbolas Centered at the Origin 563
c. x2 10 y2 12 1 2. If the eccentricity is close to 1, then the hyperbola does not open very wide. Explain why this is true. 564 Chapter l Hyperbolas
Problem 6 Wrap Up Determine the following for each hyperbola. l Whether the major axis is horizontal or vertical l The coordinates of the vertices l The coordinates of the co-vertices l The equations of the asymptotes l The coordinates of the foci l The eccentricity Then, graph the hyperbola. 1. y 2 144 x2 25 1 Lesson.1 l Hyperbolas Centered at the Origin 565
2. x 2 400 y2 225 1 566 Chapter l Hyperbolas
3. Complete the following table. Hyperbola Centered at Origin y y Graph c a x a c x Equation of Hyperbola Major Axis Coordinates of Center Coordinates of Vertices Coordinates of Co-Vertices Coordinates of Foci Asymptotes Length of Major Axis Length of Semi-Major Axis Length of Minor Axis Length of Semi-Minor Axis Be prepared to share your methods and solutions. Lesson.1 l Hyperbolas Centered at the Origin 567
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.2 And Then a Step to the Right! Hyperbolas Not Centered at the Origin Objectives In this lesson you will: l Write equations that demonstrate transformations of location or shape of hyperbola. l Graph hyperbolas through the use of transformations. l Determine key characteristics of hyperbolas from graphs or equations. Problem 1 Transformations of Hyperbolas 1. Write an equation to represent each ellipse. a. An ellipse with its center at the origin, a major axis with a length of 10 and a minor axis with a length of 6. b. An ellipse with its center at (1, 5), a major axis with a length of 10 and a minor axis with a length of 6. c. An ellipse with its center at ( 3, 8), a major axis with a length of 10 and a minor axis with a length of 6. d. An ellipse with its center at ( 4, 2), a major axis with a length of 10 and a minor axis with a length of 6. Lesson.2 l Hyperbolas Not Centered at the Origin 569
2. What is the relationship between the center of an ellipse and its equation? 3. Describe the transformations applied to the ellipse x2 a y2 1 to result 2 2 b in each ellipse. Then identify the center of the transformed ellipse. (x 1)2 y2 a. a 2 b 1 2 b. x2 ( y 7)2 2 a b 1 2 c. (x 6)2 ( y 3)2 2 a b 1 2 Take Note The process used to graph a hyperbola using a graphing calculator is similar to the process used to graph an ellipse. Solve the equation for y. Enter two separate equations for y 1 and y 2 representing the upper and lower portions of the hyperbola. 570 Chapter l Hyperbolas
Ellipses and hyperbolas have many similar characteristics. Let s see how they apply to transformations. 4. With a calculator, graph the hyperbolas x2 9 (x 8) 2 9 y2 4 1, and (x 15)2 9 y2 4 1. y2 4 1, (x 10)2 9 y2 1, 4 5. What do you notice about the shape of each hyperbola? Lesson.2 l Hyperbolas Not Centered at the Origin 571
6. Describe the transformations applied to the hyperbola x2 y2 1 to result in 9 4 each hyperbola from Question 4. 7. What is the center of each hyperbola from Question 4? 8. What is the relationship between the center of each hyperbola and its equation? 572 Chapter l Hyperbolas
9. Graph the hyperbolas x2 9 and x2 9 ( y 11)2 4 1. y2 4 1, x2 9 ( y 6)2 4 1, x2 9 ( y 7)2 1, 4 10. What do you notice about the shape of each hyperbola? Lesson.2 l Hyperbolas Not Centered at the Origin 573
11. Describe the transformations applied to the hyperbola x2 y2 1 to result in 9 4 each hyperbola from Question 9. 12. What is the center of each hyperbola from Question 9?. What is the relationship between the center of each hyperbola and its equation? 574 Chapter l Hyperbolas
14. Graph the hyperbolas x2 9 (x 9) 2 9 ( y 3)2 4 y2 (x 1, 4)2 ( y 5)2 1, 4 9 4 (x 8)2 1, and ( y 5)2 1. 9 4 15. What do you notice about the shape of each hyperbola? Lesson.2 l Hyperbolas Not Centered at the Origin 575
16. Describe the transformations applied to the hyperbola x2 y2 1 to result in 9 4 each hyperbola from Question 14. 17. What is the center of each hyperbola from Question 14? 18. What is the relationship between the center of each hyperbola and its equation? 576 Chapter l Hyperbolas
19. Identify the center of each hyperbola. Then, graph each hyperbola. (x 2)2 a. ( y 1)2 1 16 36 b. ( y 1)2 (x 2)2 1 36 16 Lesson.2 l Hyperbolas Not Centered at the Origin 577
c. (x 1)2 36 ( y 2)2 1 49 Problem 2 Connecting Equations, Key Characteristics, and Graphs Determine the following for each hyperbola. l The coordinates of the center l Whether the major axis is horizontal or vertical l The coordinates of the vertices l The coordinates of the co-vertices l The asymptotes l The coordinates of the foci l The eccentricity Then, graph and label the hyperbola. 578 Chapter l Hyperbolas
1. ( y 3)2 144 (x 5)2 1 25 Lesson.2 l Hyperbolas Not Centered at the Origin 579
2. (x 12)2 36 ( y 6)2 1 64 580 Chapter l Hyperbolas
3. ( y 10)2 (x 16)2 1 225 64 Lesson.2 l Hyperbolas Not Centered at the Origin 581
4. Complete the following table. Hyperbola y y Graph (h, k) c a x c a (h, k) x Equation of Ellipse Major Axis Coordinates of Center Coordinates of Vertices Coordinates of Co-Vertices Coordinates of Foci Asymptotes Length of Major Axis Length of Semi-Major Axis Length of Minor Axis Length of Semi-Minor Axis Be prepared to share your methods and solutions. 582 Chapter l Hyperbolas
.3 Graphs, Equations, and Key Characteristics of Hyperbolas Forms of Hyperbolas Objectives In this lesson you will: l Write equations of hyperbolas from key characteristics or graphs of hyperbolas. l Graph hyperbolas from key characteristics or equations of hyperbolas. l Extract key characteristics of hyperbolas from graphs and equations of hyperbolas. l Convert between equations of hyperbolas in general and standard form. Problem 1 Writing Equations of Hyperbolas Write an equation in standard form for each hyperbola. Then, graph the hyperbola. 1. A hyperbola with a center at (0, 0), one vertex at (3, 0), and one focus at (5, 0). Lesson.3 l Forms of Hyperbolas 583
2. A hyperbola with a center at (3, 2), one vertex at (3, 10), and one focus at (3, 12). 3. A hyperbola with vertices at (5, 1) and (5, 7) and eccentricity of 3 2. 584 Chapter l Hyperbolas
4. A hyperbola centered at (3, 0), one vertex at (9, 0) and asymptotes y 4 x 4 and y 4 x 4. 3 3 5. A hyperbola with center at (2, 2), the length of the vertical minor axis is 18 units and the length of the major axis is 11 units. Lesson.3 l Forms of Hyperbolas 585
6. A hyperbola with one vertex at (5, 0) and asymptotes y 12 x and y 12 5 5 x. 586 Chapter l Hyperbolas
Problem 2 Standard and General Form 1. Write an equation in standard form for each hyperbola. Then, determine the following for each hyperbola. l Whether the major axis is horizontal or vertical l The coordinates of the center l The coordinates of the vertices l The coordinates of the co-vertices l The coordinates of the foci l The asymptotes l The eccentricity Finally, graph and label the hyperbola. a. 4x 2 16y 2 64 0 Lesson.3 l Forms of Hyperbolas 587
b. 4x 2 9y 2 8x 36y 4 0 588 Chapter l Hyperbolas
c. 64x 2 225y 2 512x 50y 399 0 Lesson.3 l Forms of Hyperbolas 589
2. Write an equation in general form for each hyperbola. (x 4)2 y2 a. 100 121 1 b. (y 7)2 (x 1)2 25 200 1 c. (x 5)2 (y 3)2 1 81 1 590 Chapter l Hyperbolas
Problem 3 Writing Equations Using Graphs Write an equation in standard form for each hyperbola. 1. y 8 6 4 2 6 4 2 2 2 4 6 8 10 x 4 6 8 Lesson.3 l Forms of Hyperbolas 591
2. y 8 6 4 2 12 10 8 6 4 2 2 2 4 x 4 6 8 592 Chapter l Hyperbolas
3. 5 4 3 2 1 y 20 16 12 8 4 4 8 12 1 x 2 3 Be prepared to share your methods and solutions. Lesson.3 l Forms of Hyperbolas 593
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.4 Floodlights and Flashlights and LORAN Hyperbolas and Problem Solving Objectives In this lesson you will: l Model real-world situations using equations and graphs of hyperbolas. l Use key characteristics, graphs, and equations of hyperbolas to determine locations and dimensions to solve real-world situations. Problem 1 Floodlights and Flashlights The curve of the surface of a floodlight is designed in the shape of a hyperbola with the light source positioned at one focus. When light rays originating at one focus reflect off the surface of the hyperbola, the light travels in a path along the line from the other focus to the intersection point as shown. Back (reflective side) of floodlight y 8 6 4 Light Source 10 5 O 2 2 4 6 8 5 The Illuminate brand of floodlights offers one model whose surface can be represented by the equation x2 9 10 x y2 1, with dimensions in inches. 16 1. Draw additional light rays in the diagram to demonstrate how the floodlight spreads light. Lesson.4 l Hyperbolas and Problem Solving 595
2. Determine the coordinates of the vertices and co-vertices of the hyperbola. 3. Determine the coordinates of the foci. 4. How far should the light source be placed from the vertex of the hyperbolic surface to create a floodlight? Explain your answer. 596 Chapter l Hyperbolas
Problem 2 LORAN LORAN (LOng RAnge Navigation System) is one method used to determine the location of ships at sea. LORAN consists of a network of land-based transmitters. Using a ship s distance from three of these transmitters, the ship s exact location can be calculated. Plotted on a coordinate plane with one unit equaling one mile, the coordinates of transmitter A are (0, 126), the coordinates of transmitter B are (0, 0), and the coordinates of transmitter C are (112, 0). A ship is located 104 miles from transmitter A, 50 miles from transmitter B, and 78 miles from transmitter C. Specifically, LORAN determines the equation of a hyperbola with foci at transmitters A and B passing through the location of the ship and the equation of a hyperbola with foci at transmitters B and C passing through the location of the ship. The ship s location is calculated by determining the intersection of the two hyperbolas. 1. Plot points representing each transmitter in the grid. 150 5 120 105 90 75 60 y 45 30 15 0 x 0 15 30 45 60 75 90 105 120 5 150 2. Determine the equation of a hyperbola with foci at transmitters A and B passing through the location of the ship by performing the following: a. Determine the center of the hyperbola. b. Determine the difference of the distances from the ship to the foci. Lesson.4 l Hyperbolas and Problem Solving 597
c. What other distance of the hyperbola is this equal to? d. Determine the distance between the center and a co-vertex. 3. Write the equation of the hyperbola with foci at transmitters A and B passing through the location of the ship. 4. Sketch the graph of the hyperbola in the coordinate plane in Question 1. 5. Determine the equation of a hyperbola with foci at transmitters B and C passing through the location of the ship by performing the following: a. Determine the center of the hyperbola. b. Determine the difference of the distances from the ship to the foci. c. What other distance of the hyperbola is this equal to? 598 Chapter l Hyperbolas
d. Determine the distance between the center and a co-vertex. 6. Write the equation of the hyperbola with foci at transmitters B and C passing through the location of the ship. 7. Sketch the graph of the hyperbola in the coordinate plane in Question 1. 8. How many times do the two hyperbolas intersect? 9. Use a graphing calculator to determine the coordinates of each point of intersection. 10. Which intersection point represents the actual position of the ship? Explain how you determined your answer. Be prepared to share your methods and solutions. Lesson.4 l Hyperbolas and Problem Solving 599
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