Graph and Write Equations of Hyperbolas

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TEKS 9.5 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Hperbolas Before You graphed and wrote equations of parabolas, circles, and ellipses. Now You will graph and write equations of hperbolas. Wh?

Ke Vocabular hperbola foci vertices transverse ais center IDENTIFY AXES If the -term in the equation of a hperbola is positive, the transverse ais lies on the -ais.

1 TEKS 9.5 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Hperbolas Before You graphed and wrote equations of parabolas, circles, and ellipses. Now You will graph and write equations of hperbolas. Wh? So ou can model curved mirrors, as in Eample 3. Ke Vocabular hperbola foci vertices transverse ais center IDENTIFY AXES If the -term in the equation of a hperbola is positive, the transverse ais lies on the -ais. If the -term is positive, the transverse ais lies on the -ais. Recall that an ellipse is the set of all points P in a plane such that the sum of the distances between P and two fied points (the foci) is a constant. A hperbola is the set of all points P such that the difference of the distances between P and two fied points, again called the foci, is a constant. The line through the foci intersects the d 2 2 d 1 5 constant hperbola at the two vertices. The transverse ais joins the vertices. Its midpoint is the hperbola s center. A hperbola has two branches, and has two asmptotes that contain the diagonals of a rectangle centered at the hperbola s center, as shown. verte (2a, 0) (2c, 0) transverse ais (0, b) (0, 2b) verte (a, 0) (c, 0) Hperbola with horizontal transverse ais (2b, 0) (b, 0) transverse ais (0, c) (0, 2c) verte (0, a) verte (0, 2a) Hperbola with vertical transverse ais a b a b d 2 P d 1 KEY CONCEPT For Your Notebook Standard Equation of a Hperbola with Center at the Origin Equation Transverse Ais Asmptotes Vertices a b Horizontal 56b a (6a, 0) a b Vertical 56a b (0, 6a) The foci lie on the transverse ais, c units from the center, where c 2 5 a 2 1 b Chapter 9 Quadratic Relations and Conic Sections

E XAMPLE 1 Graph an equation of a hperbola Graph 25 2 4 5 100. Identif the vertices, foci, and asmptotes of the hperbola. Solution STEP 1 Rewrite the equation in standard form.

2 E XAMPLE 1 Graph an equation of a hperbola Graph Identif the vertices, foci, and asmptotes of the hperbola. Solution STEP 1 Rewrite the equation in standard form Write original equation Divide each side b Simplif. STEP 2 Identif the vertices, foci, and asmptotes. Note that a and b , so a 5 2 and b 5 5. The -term is positive, so the transverse ais is vertical and the vertices are at (0, 62). Find the foci. c 2 5 a 2 1 b , so c 5 Ï 29 The foci are at (0, 6Ï 29) ø (0, 65.4). The asmptotes are 56 a b, or (0, 29) SOLVE FOR Y To plot points on the hperbola, solve its equation for to obtain 562Î Then 25 make a table of values. STEP 3 Draw the hperbola. First draw a rectangle centered at the origin that is 2a 5 4 units high and 2b 5 10 units wide. The asmptotes pass through opposite corners of the rectangle. Then, draw the hperbola passing through the vertices and approaching the asmptotes. at classzone.com (25, 0) 1 (0, 2) (5, 0) 3 (0, 22) (0, 2 29) E XAMPLE 2 Write an equation of a hperbola Write an equation of the hperbola with foci at (24, 0) and (4, 0) and vertices at (23, 0) and (3, 0). Solution The foci and vertices lie on the -ais equidistant from the origin, so the transverse ais is horizontal and the center is the origin. The foci are each 4 units from the center, so c 5 4. The vertices are each 3 units from the center, so a 5 3. Because c 2 5 a 2 1 b 2, ou have b 2 5 c 2 2 a 2. Find b 2. (23, 0) (24, 0) (0, 7) 2 2 (3, 0) (4, 0) b 2 5 c 2 2 a (0, 2 7) Because the transverse ais is horizontal, the standard form of the equation is as follows: Substitute 3 for a and 7 for b Simplif. 9.5 Graph and Write Equations of Hperbolas 643

3 GUIDED PRACTICE for Eamples 1 and 2 Graph the equation. Identif the vertices, foci, and asmptotes of the hperbola Write an equation of the hperbola with the given foci and vertices. 4. Foci: (23, 0), (3, 0) 5. Foci: (0, 210), (0, 10) Vertices: (21, 0), (1, 0) Vertices: (0, 26), (0, 6) E XAMPLE 3 TAKS REASONING: Multi-Step Problem PHOTOGRAPHY You can take panoramic photographs using a hperbolic mirror. Light ras heading toward the behind the mirror are reflected to a camera positioned at the other as shown. After a photograph is taken, computers can unwarp the distorted image into a 3608 view. Write an equation for the cross section of the mirror. The mirror is 6 centimeters wide. How tall is it? Solution STEP 1 From the diagram, a and c To write an equation, find b 2. b 2 5 c 2 2 a ø 5.50 c Because the transverse ais is vertical, the standard form of the equation for the cross section of the mirror is as follows: , or STEP 2 Find the -coordinate at the mirror s bottom edge. Because the mirror is 6 centimeters wide, substitute 5 3 into the equation and solve Substitute 3 for. AVOID ERRORS The mirror is below the -ais, so choose the negative square root. ø Solve for. ø Solve for. c So, the mirror has a height of (24.56) centimeters. GUIDED PRACTICE for Eample 3 6. WHAT IF? In Eample 3, suppose that the mirror remains 6 centimeters wide, but that a 5 3 centimeters and c 5 5 centimeters. How tall is the mirror? 644 Chapter 9 Quadratic Relations and Conic Sections

9.5 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 13, 23, and 41 5 TAKS PRACTICE AND REASONING Es. 15, 26, 33, 35, 43, 45, and 46 5 MULTIPLE REPRESENTATIONS E. 42 1.

WRITING Compare the definitions of an ellipse and a hperbola. EXAMPLE 1 on p. 643 for Es. 3 17 GRAPHING Graph the equation. Identif the vertices, foci, and asmptotes of the hperbola. 3. 25 2 2 4 5 1 4.

TAKS REASONING What are the foci of the hperbola with equation 45 2 200 5 1800? A (62Ï 10, 0) B (0, 62Ï 10) C (67, 0) D (0, 67) ERROR ANALYSIS Describe and correct the error in graphing the equation.

4 9.5 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 13, 23, and 41 5 TAKS PRACTICE AND REASONING Es. 15, 26, 33, 35, 43, 45, and 46 5 MULTIPLE REPRESENTATIONS E VOCABULARY Cop and complete: The points (22, 0) and (2, 0) in the graph at the right are the? of the hperbola. The line segment joining these two points is the? WRITING Compare the definitions of an ellipse and a hperbola. EXAMPLE 1 on p. 643 for Es GRAPHING Graph the equation. Identif the vertices, foci, and asmptotes of the hperbola TAKS REASONING What are the foci of the hperbola with equation ? A (62Ï 10, 0) B (0, 62Ï 10) C (67, 0) D (0, 67) ERROR ANALYSIS Describe and correct the error in graphing the equation EXAMPLE 2 on p. 643 for Es WRITING EQUATIONS Write an equation of the hperbola with the given foci and vertices. 18. Foci: (0, 24), (0, 4) 19. Foci: (26, 0), (6, 0) Vertices: (0, 22), (0, 2) Vertices: (22, 0), (2, 0) 20. Foci: (25, 0), (5, 0) 21. Foci: (0, 212), (0, 12) Vertices: (21, 0), (1, 0) Vertices: (0, 27), (0, 7) 22. Foci: (210, 0), (10, 0) 23. Foci: (0, 24Ï 5 ), (0, 4Ï 5 ) Vertices: (25Ï 3, 0), (5Ï 3, 0) Vertices: (0, 24), (0, 4) 24. Foci: (0, 23), (0, 3) 25. Foci: (23Ï 6, 0), (3Ï 6, 0) Vertices: (0, 22Ï 2 ), (0, 2Ï 2 ) Vertices: (22, 0), (2, 0) 9.5 Graph and Write Equations of Hperbolas 645

5 26. TAKS REASONING What is an equation of the hperbola with foci at (0, 26Ï 3 ) and (0, 6Ï 3 ) and with vertices at (0, 28) and (0, 8)? A B C D GRAPHING In Eercises 27 32, the equations of parabolas, circles, ellipses, and hperbolas are given. Graph the equation TAKS REASONING Describe the effects of the indicated change on the shape of the hperbola and on the locations of the vertices and foci. a ; change 36 to 4 b ; change 4 to GRAPHING CALCULATOR Graph each hperbola using a graphing calculator. Tell what two functions ou entered into the calculator. a b c TAKS REASONING Give equations of three hperbolas with horizontal transverse aes and asmptotes 562. Compare the hperbolas. 36. REASONING Use the diagram at the right to show that d 2 2 d 1 5 2a. (Hint: d 2 2 d 1 is constant, so choose a convenient location for (, ).) 37. CHALLENGE Using the distance formula and the definition of a hperbola, write an equation in standard form of the hperbola with foci at (62, 0) if the difference in the distances from a point (, ) on the hperbola to the foci is 2. (2c, 0) (2a, 0) (a, 0) (c, 0) (, ) d 2 d 1 PROBLEM SOLVING EXAMPLE 3 on p. 644 for Es TELESCOPES A satellite is carring a telescope that has a hperbolic mirror for which a 5 33 and c 5 56 (in centimeters). Write an equation for the cross section of the mirror if the transverse ais is horizontal. 39. SPINNING CUBE The outline of a cube spinning around an ais through a pair of opposite corners contains a portion of a hperbola, as shown. The coordinates given represent a verte and a of the hperbola for a cube that measures 1 unit on each edge. Write an equation that models this hperbola. 1 Ï 2 2, 02 1 Ï 3 2, Chapter 9 5 WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

40. SUN S SHADOW Each da, ecept at the fall and spring equinoes, the tip of the shadow of a vertical pole traces a branch of a hperbola across the ground.

Write an equation of the hperbola with center at the origin that models the June 21 path, given that a 5 13.3 meters and c 5 28.2 meters. b.

Louis Science Center has a hperbolic cross section with the dimensions shown. a. Suppose a coordinate grid is overlaid on the diagram with its origin at O, the center of the narrowest part of the roof.

MULTIPLE REPRESENTATIONS A circular walkwa is to be built around a statue in a park. There is enough concrete available for the walkwa to have an area of 600 square feet. a. Writing an Equation Let the inside and outside radii of the walkwa be feet and feet, respectivel.

Drawing a Graph Graph the equation from part (a). What portion of the graph represents solutions that make sense in this situation? d.

6 40. SUN S SHADOW Each da, ecept at the fall and spring equinoes, the tip of the shadow of a vertical pole traces a branch of a hperbola across the ground. The diagram shows shadow paths for a 20 meter tall flagpole in Dallas, Teas. N Dec 21 Jun 21 N Feb 27 Sep 1 (13.3, 0) Focus (28.2, 0) Focus (4.1, 0) (25.0, 0) a. Write an equation of the hperbola with center at the origin that models the June 21 path, given that a meters and c meters. b. Write an equation of the hperbola with center at the origin that models the September 1 path, given that a meters and c meters. 41. MULTI-STEP PROBLEM The roof of the St. Louis Science Center has a hperbolic cross section with the dimensions shown. a. Suppose a coordinate grid is overlaid on the diagram with its origin at O, the center of the narrowest part of the roof. What are the coordinates of the points at A and B? b. Use our answers from part (a) to write an equation that models the cross section. c. Find the total height h of the roof. 42. MULTIPLE REPRESENTATIONS A circular walkwa is to be built around a statue in a park. There is enough concrete available for the walkwa to have an area of 600 square feet. a. Writing an Equation Let the inside and outside radii of the walkwa be feet and feet, respectivel. Draw a diagram of the situation. Then write an equation relating and. b. Making a Table Give four possible pairs of dimensions and that satisf the equation from part (a). c. Drawing a Graph Graph the equation from part (a). What portion of the graph represents solutions that make sense in this situation? d. Reasoning How does the width of the walkwa,, change as both and increase? Eplain wh this makes sense. 43. TAKS REASONING Two stones dropped at the same time into still water produce circular ripples whose intersection points form hperbolas with foci where the stones hit the water. The graph shows one hperbola formed b stones dropped 12 feet apart with ripples at 1 foot intervals. a. Write an equation of this hperbola. b. Use the definition of a hperbola to eplain wh the graph shown is a hperbola. (Hint: Eamine the distances from each intersection point to the foci.) (26, 0) (24, 0) (4, 0) (6, 0) 9.5 Graph and Write Equations of Hperbolas 647

44. CHALLENGE Two microphones placed 1 mile apart record the bugling of a bull elk. Microphone A receives the sound 2 seconds after microphone B. Sound travels at 1100 feet per second.

7 44. CHALLENGE Two microphones placed 1 mile apart record the bugling of a bull elk. Microphone A receives the sound 2 seconds after microphone B. Sound travels at 1100 feet per second. Is this enough information to determine where the elk is located? If so, give the location. If not, eplain wh not. MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 2.3; TAKS Workbook 45. TAKS PRACTICE Which equation does the graph represent? TAKS Obj. 1 A B C D (26, 22) (22, 4) REVIEW Lesson 9.1; TAKS Workbook 46. TAKS PRACTICE The endpoints of a diameter of a circle are (25, 8) and (9, 23). What is the center of the circle? TAKS Obj. 7 F 1 28, G (2, 3) H 1 2, J 1 7, QUIZ for Lessons Graph the equation. Identif the vertices, co-vertices, and foci of the ellipse. (p. 634) Write an equation of the ellipse with the given characteristics and center at (0, 0). (p. 634) 4. Verte: (0, 5) 5. Verte: (10, 0) 6. Co-verte: (2Ï 15, 0) Co-verte: (24, 0) Focus: (28, 0) Focus: (0, 25) Graph the equation. Identif the vertices, foci, and asmptotes of the hperbola. (p. 642) Write an equation of the hperbola with the given foci and vertices. (p. 642) 10. Foci: (25, 0), (5, 0) 11. Foci: (0, 23), (0, 3) 12. Foci: (23Ï 6, 0), (3Ï 6, 0) Vertices: (22, 0), (2, 0) Vertices: (0, 21), (0, 1) Vertices: (23, 0), (3, 0) 13. ASTEROIDS The largest asteroid, 1 Ceres, ranges from 2.55 astronomical units to 2.98 astronomical units from the sun, which is located at one of the asteroid s elliptical orbit. Find a and c. Then write an equation of the orbit of 1 Ceres. (p. 634) 648 EXTRA PRACTICE for Lesson 9.5, p ONLINE QUIZ at classzone.com

Investigating g Algebra ACTIVITY Use before Lesson 9.6 Algebra classzone.com 9.

A MATERIALS flashlight graph paper QUESTION How do a plane and a double-napped cone intersect to form different conic sections?

the intersection of a plane and a double-napped cone, as shown below.

On a piece of graph paper, draw - and -aes to make a coordinate plane. Then tape the paper to a wall.

the coordinate plane. Trace the circle on the graph paper.

STEP 3 Model an ellipse Tilt the flashlight, and aim it at the paper to form an ellipse with a vertical major ais and center

8 Investigating g Algebra ACTIVITY Use before Lesson 9.6 Algebra classzone.com 9.6 Eploring Intersections of TEKS Planes and Cones 2A.5.A MATERIALS flashlight graph paper QUESTION How do a plane and a double-napped cone intersect to form different conic sections? The reason that parabolas, circles, ellipses, and hperbolas are called conics or conic sections is that each can be formed b the intersection of a plane and a double-napped cone, as shown below. Circle Ellipse Parabola Hperbola E XPLORE Find an equation of a conic STEP 1 Draw aes Work in a group. On a piece of graph paper, draw - and -aes to make a coordinate plane. Then tape the paper to a wall. STEP 2 Model a circle Aim a flashlight perpendicular to the paper so that the light forms a circle centered on the origin of the coordinate plane. Trace the circle on the graph paper. Find the circle s radius, and use it to write the standard form of the circle s equation. STEP 3 Model an ellipse Tilt the flashlight, and aim it at the paper to form an ellipse with a vertical major ais and center at the origin. Trace the ellipse and write the standard form of its equation. DRAW CONCLUSIONS Use our observations to complete these eercises 1. Compare the equations for our circle and for our ellipse with the equations of other groups. Are our equations all the same? Wh or wh not? 2. Refer to the diagram of a hperbola to eplain how ou can orient the flashlight beam to form a branch of a hperbola on the wall. 9.6 Translate and Classif Conic Sections 649

8.5 Graph and Write Equations of Hyperbolas p.518 What are the parts of a hyperbola? What are the standard form equations of a hyperbola?

8.5 Graph and Write Equations of Hyperbolas p.518 What are the parts of a hyperbola? What are the standard form equations of a hyperbola? How do you know which way it opens? Given a & b, how do you find