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Section 9.2 The Hyperbola Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: • • • • • Locate a hyperbola’s vertices and foci. Write equations of hyperbolas in standard form. Graph hyperbolas centered at the origin. Graph hyperbolas not centered at the origin. Solve applied problems involving hyperbolas. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Definition of a Hyperbola A hyperbola is the set of points in a plane the difference of whose distances from two fixed points, called foci, is constant. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Two Branches of a Hyperbola The line through the foci intersects the hyperbola at two points, called the vertices. The line segment that joins the vertices is the transverse axis. The midpoint of the transverse axis is the center of the hyperbola. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Standard Forms of the Equations of a Hyperbola The standard form of the equation of a hyperbola with center at the origin is x 2 y 2 y 2 x2 2 1 or 2 2 1. 2 a b a b Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Standard Forms of the Equations of a Hyperbola (continued) The vertices are a units from the center and the foci are c units from the center. For both equations, b2 = c2 – a2. Equivalently, c2 = a2 + b2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Finding Vertices and Foci from a Hyperbola’s Equation Find the vertices and locate the foci for the hyperbola with the given equation: x 2 y 2 1. 25 16 a 2 25, a 5. The vertices are (–5, 0) and (5, 0). c2 a 2 b2 c 2 25 16 41 c 41 The foci are at 41,0 and 41,0 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Finding Vertices and Foci from a Hyperbola’s Equation (continued) Find the vertices and locate the foci for the hyperbola with the given equation: 2 2 x y 1 25 16 The vertices are (–5, 0) and (5, 0). The foci are at 41,0 and 41,0 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Finding Vertices and Foci from a Hyperbola’s Equation Find the vertices and locate the foci for the hyperbola 2 2 y x with the given equation: 1. 25 16 a 25, a 5. The vertices are (0, –5) and (0, 5). 2 c2 a 2 b2 c 2 25 16 41 c 41 The foci are at 0, 41 and 0, 41 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Finding the Equation of a Hyperbola from Its Foci and Vertices (continued) Find the vertices and locate the foci for the hypberbola with the given equation: 2 2 y x 1 25 16 The vertices are (0, –5) and (0, 5). The foci are at 0, 41 and 0, 41 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Finding the Equation of a Hyperbola from Its Foci and Vertices Find the standard form of the equation of a hyperbola with foci at (0, –5) and (0, 5) and vertices (0, –3) and (0, 3). Because the foci are located at (0, –5) and (0, 5), the transverse axis lies on the y-axis. Thus, the form of the equation is y 2 x 2 2 1. 2 a b The distance from the center to either vertex is 3, so a = 3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Finding the Equation of a Hyperbola from Its Foci and Vertices Find the standard form of the equation of a hyperbola with foci at (0, –5) and (0, 5) and vertices (0, –3) and (0, 3). The distance from the center to either focus is 5. Thus, c = 5. The equation is c2 a 2 b2 y 2 x2 1 2 25 9 b 9 16 b 2 16 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 The Asymptotes of a Hyperbola Centered at the Origin Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 The Asymptotes of a Hyperbola Centered at the Origin (continued) As x and y get larger, the two branches of the graph of a hyperbola approach a pair of intersecting straight lines, called asymptotes. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Graphing Hyperbolas Centered at the Origin Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Graphing a Hyperbola 2 2 x y Graph and locate the foci: 1. 36 9 What are the equations of the asymptotes? Step 1 Locate the vertices. x2 y 2 The given equation is in the form 2 2 1 a b 2 2 with a = 36 and b = 9. a2 = 36, a = 6. Thus, the vertices are (–6, 0) and (6, 0). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Graphing a Hyperbola (continued) 2 2 x y Graph and locate the foci: 1. 36 9 Step 2 Draw a rectangle a2 = 36, a = 6. b2 = 9, b = 3. The rectangle passes through the points (–6, 0) and (6, 0) and (0, –3) and (0, 3). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Graphing a Hyperbola (continued) 2 2 x y Graph and locate the foci: 1. 36 9 Step 3 Draw extended diagonals for the rectangle to obtain the asymptotes. b 3 1 a 6 2 The equations for the asymptotes 1 1 are y x and y x. 2 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Graphing a Hyperbola (continued) 2 2 x y Graph and locate the foci: 1. 36 9 Step 4 Draw the two branches of the hyperbola starting at each vertex and approaching the asymptotes. c 2 a 2 b 2 c 2 36 9 45 c 45 6.4 The foci are at 45,0 and or (–6.4, 0) and (6.4, 0). 45,0 , Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Graphing a Hyperbola (continued) 2 2 Graph and locate the foci: x y 1. 36 9 What are the asymptotes? The foci are 45,0 and 45,0 , or (–6.7, 0) and (6.7, 0). The equations of the asymptotes 1 1 are y x and y x. 2 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Translations of Hyperbolas – Standard Forms of Equations of Hyperbolas Centered at (h, k) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Translations of Hyperbolas – Standard Forms of Equations of Hyperbolas Centered at (h, k) (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Graphing a Hyperbola Centered at (h, k) Graph: 4 x2 24 x 9 y 2 90 y 153 0 We begin by completing the square on x and y. 4 x 24 x 9 y 90 y 153 0 2 2 (4 x 24 x) (9 y 90 y) 153 2 4 x2 6 x 2 9 y 2 10 y 153 4( x2 6 x 9) 9( y 2 10 y 25) 153 36 225 4( x 3)2 9( y 5)2 36 2 2 2 2 4( x 3) 9( y 5) 36 ( x 3) ( y 5) 1 36 36 36 9 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Graphing a Hyperbola Centered at (h, k) (continued) Graph: 4 x2 24 x 9 y 2 90 y 153 0 ( x 3)2 ( y 5) 2 ( y 5)2 ( x 3) 2 1 1 9 4 4 9 We will graph ( y 5)2 ( x 3) 2 1 4 9 The center of the graph is (3, –5). Step 1 Locate the vertices. 2 a 4, a 2. The vertices are 2 units above and below the center. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 Step 1 (cont) Locate the vertices. The center is (3, –5). The vertices are 2 units above and 2 units below the center. 2 units above (3, –5 + 2) = (3, –3) 2 units below (3, –5 – 2) = (3, –7) The vertices are (3, –3) and (3, –7). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 Step 2 Draw a rectangle. The rectangle passes through points that are 2 units above and below the center - these points are the vertices, (3, –3) and (3, –7). The rectangle passes through points that are 3 units to the right and left of the center. 3 units right (3 + 3, –5) = (6, –5) 3 units left (3 – 3, –5) = (0, –5) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 Step 2 (cont) Draw a rectangle. The rectangle passes through (3, –3), (3, –7), (0, –5), and (6, –5). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 Step 3 Draw extended diagonals of the rectangle to obtain the asymptotes. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 Step 3 (cont) Draw extended diagonals of the rectangle to obtain the asymptotes. The asymptotes for the unshifted 2 hyperbola are y x. 3 The asymptotes for the shifted hyperbola are Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 Step 3 (cont) 2 The asymptotes for the unshifted hyperbola are y x. 3 The asymptotes for the shifted hyperbola are 2 y 5 ( x 3). 3 2 2 2 y 5 ( x 3) y5 x2 y x7 3 3 3 2 y 5 ( x 3) 3 2 y5 x2 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 y x3 3 30 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 Step 4 Draw the two branches of the hyperbola by starting at each vertex and approaching the asymptotes. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 The foci are at (3, –5 – c) and (3, –5 + c). c2 a 2 b2 c 4 9 13 2 c 13 The foci are 3, 5 13 and 3, 5 13 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 32 Example: Graphing a Hyperbola Centered at (h, k) (continued) 2 2 ( y 5) ( x 3) Graph: 1 4 9 The center is (3, –5). The vertices are (3, –3) and (3, –7). The foci are 3, 5 13 and 3, 5 13 . The asymptotes are 2 y x7 3 2 and y x 3. 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 33 Example: An Application Involving Hyperbolas An explosion is recorded by two microphones that are 2 miles apart. Microphone M1 receives the sound 3 seconds before microphone M2. Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones. We begin by putting the microphones in a coordinate system. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 34 Example: An Application Involving Hyperbolas (continued) We know that M1 received the sound 3 seconds before M2. Because sound travels at 1100 feet per second, the difference between the distance from P to M1 and the distance from P to M2 is 3300 feet. The set of all points P (or locations of the explosion) satisfying these conditions fits the definition of a hyperbola, with microphones M1 and M2 at the foci. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 35 Example: An Application Involving Hyperbolas (continued) We will use the standard form of the hyperbola’s equation. P(x, y), the explosion point, lies on this hyperbola. x2 y 2 2 1 2 a b The differences between the distances is 3300 feet. Thus, 2a = 3300 and a = 1650. x2 y2 2 1 2 1650 b x2 y2 2 1 2,722,500 b Copyright © 2014, 2010, 2007 Pearson Education, Inc. 36 Example: An Application Involving Hyperbolas (continued) The distance from the center, (0, 0), to either focus (–5280, 0) or (5280, 0) is 5280. Thus, c = 5280. c2 a 2 b2 52802 16502 b 2 b2 52802 16502 25,155,900 The equation of the hyperbola with a microphone at each focus is x2 y2 1. 2,722,500 25,155,900 We can conclude that the explosion occurred somewhere on the right branch (the branch closer to M1) of the hyperbola given by this equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 37