Chapter 5: The Hyperbola
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1 Chapter 5: The Hyperbola SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
2 Chapter 5: The Hyperbola Lecture 17: Introduction to Hyperbola Lecture 18: The Parts of the Graph of the Hyperbola Lecture 19: Converting General Equation of a Hyperbola to Standard Equation and Vice-Versa Lecture 0: Graphing a Hyperbola with Center at the Origin C (0, 0) Lecture 1: Graphing a Hyperbola with Center at C (h, k)
3 Lecture 17: The Introduction to Hyperbola SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
4 A Short Recap In our introductory lesson about conic sections, how does a hyperbola formed? What are the conditions we need to satisfy?
5 Remember the Ellipse?
6 How about the Hyperbola?
![Definition of Hyperbola A hyperbola is a set of all points in a plane [P (x, y)] such that the](/docs-images/72/66622122/images/7-2.jpg "difference of the distances of each point [P (x, y)] from two fixed points ( F 1 and F ) is")
7 Definition of Hyperbola A hyperbola is a set of all points in a plane [P (x, y)] such that the difference of the distances of each point [P (x, y)] from two fixed points ( F 1 and F ) is constant.
8 Definition of Hyperbola
9 Did you know? A hyperbola can also be defined as a curve with no ends where the distance of any point from a fixed point (the focus), and the fixed line (the directrix) are always in the same ratio.
10 Did you know? The shape of the hyperbola is dependent on the value of eccentricity (e). The eccentricity of the hyperbola is e>1, as e approaches 1, the hyperbola becomes narrow.
11 Did you Ever Wonder? Did you ever wonder why it is called HYPERBOLA?
12 The Eccentricity of the Conic Section
13 Two Types of Hyperbola 1. Horizontal Hyperbola. Vertical Hyperbola
14 The Horizontal Hyperbola If the center of the hyperbola is the origin and the foci are in the x-axis, then it is a horizontal hyperbola.
15 The Horizontal Hyperbola
16 The Vertical Hyperbola If the center of the hyperbola is the origin and the foci are in the y-axis, then it is a vertical hyperbola.
17 The Vertical Hyperbola
18 Classroom Task Identify whether each of the picture which will be shown modelled a hyperbola conic section. Justify your answer.
19 Shadow of a Lamp Shade
20 Cooling Towers of Nuclear Reactors
21 Did you ever wonder? Did you ever wonder why the cooling towers of nuclear reactors was designed like that?
22 Improvised Egg Tray
23 Basketball
24 Did you ever wonder? Did you ever wonder why the basketball has hyperbolic design on it?
25 TED Ed Video: Hyperbola in Real-World Context The Sonic Boom Problem by Katerina Kouri
26 The Horizontal Hyperbola
27 Representations: Let: F, be one of the fixed points with F (-c, 0); x 1 = -c and y 1 = 0; P, be the point in the plane/ ellipse with P(x, y); x = x and y = y; F 1, be one of the fixed points with F 1 (c, 0); x 3 = c and y 3 = 0; PF 1, be the distance from P(x, y) and F 1 (c, 0); PF, be the distance from P(x, y) and F (-c, 0); and k, be the difference of the distances of PF 1 and PF which is constant.
28 Theorem: The difference of the lengths of any two sides of a triangle is less than the third side.
29 Standard Equation of the Horizontal Hyperbola The standard equation of Horizontal Hyperbola with center at the origin C (0, 0): x b y 1 a
30 Standard Equation of the Horizontal Hyperbola The standard equation of Vertical Hyperbola with center at the origin C (0, 0): y b x 1 a
31 Performance Task 16: Please download, print and answer the Let s Practice 16. Kindly work independently.
32 Lecture 18: The Parts of the Graph of a Hyperbola SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
33 Did you know? Just like an ellipse, a hyperbola has a pair of foci, a pair of vertices, and a pair of directrices.
34 Did you know? In our previous lessons, how did we define the foci, vertices, co-vertices, latera recta, and directrices?
35 Foci These are the two fixed points of the hyperbola denoted by F and F. 1
36 Vertices These are the endpoints of the traverse axis.
37 Co-Vertices These are the endpoints of the conjugate axis.
38 Directrices These are line such that the ratio of distance of the points on the hyperbola from the focus to its distance from the directrix is constant.
39 Focal Length The line segment joining the foci (F 1 and F ) and has length of c.
40 A Short Review In our previous lessons about ellipse, what is our other term for major axis?
41 Traverse Axis The line segment joining the vertices (V and V ) 1 is called the traverse axis and has length of a.
42 Conjugate Axis The line segment joining the c0-vertices (B 1 and B ) and which is a perpendicular bisector of the traverse axis is called the conjugate axis. It has length of b.
43 Asymptotes These are lines passing through the center of the hyperbola and are asymptotic to the curves.
44 The Center of Ellipse It is the intersection of the traverse and conjugate axes. Also, it is the point where the two asymptotes intersect.
45 A Short Recap What do we mean when we say that a line is asymptotic to the curve?
46 Lecture 19: Converting General Equation of a Hyperbola to Its Standard Equation and Vice-Versa SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
47 What should we expect? This section presents how to convert general form of hyperbola to its standard form and viceversa.
48 Table 5.1: Equations of Hyperbola Traverse Centerc General Form Standard Form Axis (0, 0) x-axis Ax (0, 0) y-axis Cy Cy Ax F 0 F 0 x a y a b y b x 1 1 (h, k) x-axis Ax Cy Dx Ey F 0 ( x h) a ( y k) b 1 (h, k) y-axis Cy Ax Dx Ey F 0 ( y k) a ( x h) b 1
49 Family Activity 3: Converting General Equation of a Hyperbola to Its Standard Equation and Vice-Versa SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
50 Instructions: Gather your family members and answer the problem on the activity using the prior knowledge you have learned from our previous lessons. Write your complete and neat solution on your manila paper. Please be ready to share your answer to the whole class afterwards.
51 Important Reminder: You are only allowed to communicate with your family members using sign language, body movements, facial expressions, hand gestures etc. For every word your family member will utter is equivalent to a point deduction from your total score. This is a time-pressure task. The moment the music stops you are expected that you have finished doing what is instructed your family to do.
52 How We will Evaluate your Work? Criteria Percentage Correctness 50 Neatness of Computations and Organization of Ideas 0 Communication Skills 10 Presentation and Aesthetic Consideration Behavior during the Presentation 10 10
53 Example 47: Convert the given general equation to standard form: 3x y 1
54 Final Answer The standard equation is: x y 1 4 6
55 Example 48: Convert the given general equation to standard form: 9 y 16 x 144
56 Final Answer The standard equation is: y x
57 Example 49: Convert the given general equation to standard form: y 5x 30 x 4y 46 0
58 Final Answer: The standard equation is: ( y ) ( x 3) 1 5 1
59 Example 50: Convert the given general equation to standard form: 4x 9 y 16 x 18 y 10 0
60 Final Answer The standard equation is: 4( x ) 9( y 1) 3
61 Something to think about What can you observe on the right side of the equation? What can you conclude?
62 Our Conclusion 1 about Hyperbola: Since the right side of the equation is negative and the left side is positive for all points (x, y), the graph is empty set or this hyperbola DOES NOT EXIST.
63 Example 51: Convert the given general equation to standard form: 9x y 36 x 6 y 7 0
64 Final Answer The standard equation is: 9( x ) ( y 3) 0
65 Something to think about What can you observe on the right side of the equation? What can you conclude?
66 Our Conclusion about the Hyperbola: Since the right side of the equation is zero, the graph is the point (,-3). This type of equation of a hyperbola is called a DEGENERATE HYPERBOLA.
67 Example 5: Convert the given standard equation to its standard form: ( y 4) ( x 1)
68 Final Answer The general equation is: 4 y x x 3 y 7 0
69 Example 53: Convert the given standard equation to its standard form: ( x 1) ( y 9 1) 5 1
70 Final Answer The general equation is: 5 x 9 y 50 x 18 y 09 0
71 Performance Task 17: Please download, print and answer the Let s Practice 17. Kindly work independently.
72 Lecture 0: Graphing a Hyperbola with Center at the Origin C(0, 0) SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
73 Learning Expectation: This section presents how to graph a hyperbola and how to determine the parts of hyperbola where the center is set at the origin C (0, 0).
74 Table 5.: Parts of the Hyperbola with Center at C(0, 0) Standard Equation Foci Vertices/ Endpoints of Traverse Axis Covertices/ Endpoints of Conjugate Axis Endpoints of Latera Recta Asymptotes F 1 (c, 0) F (-c, 0) V 1 (a, 0) V (-a, 0) B 1 (0, b) B (0, -b) F 1 (0, c) F (0, -c) V 1 (0, a) V (0, -a) B 1 (b, 0) B (-b, 0) 1 b y a x a b c E a b c E a b c E a b c E 4 3 1,,,, x a b y 1 b x a y c a b E c a b E c a b E c a b E,,,, x b a y
75 Example 54: Sketch and discuss the given hyperbola below: 64 y 36 x,304
76
77 Example 55: Find the equation of the hyperbola with center at (0,0), traverse axis parallel to x-axis. The length of the traverse axis is 6 units and the length of the conjugate axis is 8 units. Identify the parts of the hyperbola and sketch the graph.
78 Final Answer The standard equation of the x 9 hyperbola is: y 16 1
79
80 Performance Task 18: Please download, print and answer the Let s Practice 18. Kindly work independently.
81 Lecture 1: Graphing a Hyperbola with Center at C (h, k) SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
82 Learning Expectation: This section presents how to graph a hyperbola and how to determine the parts of hyperbola with center at C (h,k).
83 Table 5.3: Parts of a Hyperbola with Center at C (h, k) Standard Equation Foci Vertices/ Endpoints of Traverse Axis Co-vertices/ Endpoints of Conjugate Axis Endpoints of Latera Recta Asymptotes F 1 (h+c, k) F (h-c, k) V 1 (h+a, k) V (h-a, k) B 1 (h, k+b) B (h, k-b) F 1 (h, k+c) F (h, k-c) V 1 (h, k+a) V (h, k-a) B 1 (h+b, k) B (h-b, k) 1 ) ( ) ( b k y a h x a b k c h E a b k c h E a b k c h E a b k c h E 4 3 1,,,, ( h) x a b k y c k a b h E c k a b h E c k a b h E c k a b h E,,,, ( h) x b a k y 1 ) ( ) ( b h x a k y
84 Example 56: Sketch and discuss the hyperbola below: 9x 5 y 00 x 54 y 544 0
85
86 Example 57: Find the equation of the hyperbola with foci at (9, ) and (-1, ), whose length of the traverse axis is 8 units long. Identify the parts of the hyperbola and sketch the graph.
87
88 Final Answer The standard equation of the hyperbola is: ( x 4) ( y 16 ) 9 1
89 Performance Task 19: Please download, print and answer the Let s Practice 19. Kindly work independently.
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