Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 7: Conic Sections 7.1 The Parabola 7.2 The Circle and the Ellipse 7.3 The.

Slides:



Advertisements
Similar presentations
Conics Hyperbola. Conics Hyperbola Cross Section.
Advertisements

Section 11.6 – Conic Sections
Section 9.2 The Hyperbola. Overview In Section 9.1 we discussed the ellipse, one of four conic sections. Now we continue onto the hyperbola, which in.
Conic Sections MAT 182 Chapter 11
Copyright © 2011 Pearson Education, Inc. Slide
10.1 Conics and Calculus. Each conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone. CircleParabolaEllipse.
§ 10.3 The Hyperbola.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1 Homework, Page 653 Find the vertices and foci of the ellipse.
Colleen Beaudoin February,  Review: The geometric definition relies on a cone and a plane intersecting it  Algebraic definition: a set of points.
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.
Hyperbola – a set of points in a plane whose difference of the distances from two fixed points is a constant. Section 7.4 – The Hyperbola.
Hyperbolas and Rotation of Conics
Conic Sections Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Conic Sections Conic sections are plane figures formed.
10.5 Hyperbolas What you should learn: Goal1 Goal2 Graph and write equations of Hyperbolas. Identify the Vertices and Foci of the hyperbola Hyperbolas.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Section 9.2 The Hyperbola.
11.4 Hyperbolas ©2001 by R. Villar All Rights Reserved.
10.3 Hyperbolas. Circle Ellipse Parabola Hyperbola Conic Sections See video!
Section 9-5 Hyperbolas. Objectives I can write equations for hyperbolas I can graph hyperbolas I can Complete the Square to obtain Standard Format of.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 10 Conic Sections.
Precalculus Warm-Up Graph the conic. Find center, vertices, and foci.
Copyright © 2007 Pearson Education, Inc. Slide 6-1.
SECTION: 10 – 3 HYPERBOLAS WARM-UP
EXAMPLE 1 Graph the equation of a translated circle
Copyright © Cengage Learning. All rights reserved.
10.4 HYPERBOLAS. 2 Introduction The third type of conic is called a hyperbola. The definition of a hyperbola is similar to that of an ellipse. The difference.
Section 3 Chapter Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives The Hyperbola and Functions Defined by Radials Recognize.
Slide 5- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
HYPERBOLA. PARTS OF A HYPERBOLA center Focus 2 Focus 1 conjugate axis vertices The dashed lines are asymptotes for the graphs transverse axis.
Advanced Geometry Conic Sections Lesson 4
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 9 Analytic Geometry.
Precalculus Chapter Analytic Geometry 10 Hyperbolas 10.4.
Section 2 Chapter Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives The Circle and the Ellipse Find an equation of a circle.
What is a hyperbola? Do Now: Define the literary term hyperbole.
Hyberbola Conic Sections. Hyperbola The plane can intersect two nappes of the cone resulting in a hyperbola.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1 Homework, Page 641 Find the vertex, focus, directrix, and focal.
10.5 Hyperbolas p.615 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.
Slide 1- 1 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Conics, Parametric Equations, and Polar Coordinates Copyright © Cengage Learning. All rights reserved.
Hyperbola Definition: A hyperbola is a set of points in the plane such that the difference of the distances from two fixed points, called foci, is constant.
Copyright © 2011 Pearson Education, Inc. Conic Sections CHAPTER 13.1Parabolas and Circles 13.2Ellipses and Hyperbolas 13.3Nonlinear Systems of Equations.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 7: Conic Sections 7.1 The Parabola 7.2 The Circle and the Ellipse 7.3 The.
10.1 Conics and Calculus.
8.5 Graph and Write Equations of Hyperbolas
Conics A conic section is a graph that results from the intersection of a plane and a double cone.
Conics 7.3: Hyperbolas Objectives:
EXAMPLE 1 Graph the equation of a translated circle
6-3 Conic Sections: Ellipses
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
. . Graphing and Writing Equations of Hyperbolas
Hyperbolas 4.4 Chapter 10 – Conics. Hyperbolas 4.4 Chapter 10 – Conics.
Section 10.3 The Hyperbola Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.
Hyperbolas.
10.3 The Hyperbola.
6-3 Conic Sections: Ellipses
Review Circles: 1. Find the center and radius of the circle.
9.5A Graph Hyperbolas Algebra II.
Chapter 10 Conic Sections
College Algebra Sixth Edition
MATH 1330 Section 8.3.
10-5 Hyperbolas Hubarth Algebra II.
MATH 1330 Section 8.3.
MATH 1330 Section 8.3.
Geometric Definition of a Hyperbola
Chapter 10 Conic Sections.
MATH 1330 Section 8.3.
Hyperbolas Chapter 8 Section 5.
Hyperbolas.
Section 11.6 – Conic Sections
Chapter 10 Conic Sections.
Chapter 7 Analyzing Conic Sections
Presentation transcript:

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 7: Conic Sections 7.1 The Parabola 7.2 The Circle and the Ellipse 7.3 The Hyperbola 7.4 Nonlinear Systems of Equations and Inequalities

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 7.3 The Hyperbola  Given an equation of a hyperbola, complete the square, if necessary, and then find the center, the vertices, and the foci and graph the hyperbola.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Hyperbola A hyperbola is the set of all points in a plane for which the absolute value of the difference of the distances from two fixed points (the foci) is constant. The midpoint of the segment between the foci is the center of the hyperbola.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Hyperbola with Center at the Origin -- Horizontal Transverse Axis Horizontal Vertices: (–a, 0), (a, 0) Foci: (–c, 0), (c, 0) where c 2 = a 2 + b 2 Segment is the conjugate axis.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Hyperbola with Center at the Origin -- Vertical Transverse Axis Vertical Vertices: (0, –a), (0, a) Foci: (0, –c), (0, c) where c 2 = a 2 + b 2 Segment is the conjugate axis.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find an equation of the hyperbola with vertices (0,  4) and (0, 4) and foci (0,  6) and (0, 6). Solution: We know that a = 3 and c = 5. We find b 2. Vertices and foci are on the y-axis, so the transverse axis is vertical.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graphing a Hyperbola - Horizontal Transverse Axis Sketch the asymptotes: y = –(b/a)x and y = (b/a)x Draw a rectangle with vertical sides through the vertices and horizontal sides through the endpoints of the conjugate axis.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example For the hyperbola given by 9x 2  16y 2 = 144, find the vertices, the foci, and the asymptotes. Then graph the hyperbola. Solution: First, we find standard form:

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) The hyperbola has a horizontal transverse axis, so the vertices are (  4, 0) and (4, 0). Thus, the foci are (–5, 0) and (5, 0). For the standard form of the equation, we know that a 2 = 5 2, or 25, and b 2 = 4 2, or 16. We find the foci:

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) Next, we find the asymptotes: To draw the graph, sketch the asymptotes first. Draw the rectangle with horizontal sides through (0, 3) and (0, – 3) and vertical sides through (4, 0) and (–4, 0). Draw and extend the diagonals of the rectangle and these are the asymptotes. Next plot the vertices and draw the branches of the hyperbola outward from the vertices towards the asymptotes.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued)

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Hyperbola with Center at (h, k) -- Horizontal Transverse Axis Horizontal Vertices: (h – a, k), (h + a, k) Asymptotes: Foci: (h – c, k), (h + c, k) where c 2 = a 2 + b 2

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Hyperbola with Center at (h, k) -- Vertical Transverse Axis Vertical Vertices: (h, k – a), (h, k + a) Asymptotes: Foci: (h, k – c), (h, k + c) where c 2 = a 2 + b 2

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example For the hyperbola given by 4y 2  x y + 4x + 28 = 0, find the center, the vertices, and the foci. Then draw the graph. Solution: Complete the square to get standard form:

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) The center is (2, –3). Note: a = 1 and b = 2. The transverse axis is vertical, so vertices are 1 unit below and above the center: (2, –3 – 1) and (2, –3 + 1), or (2, –4) and (2, –2).

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) We know c 2 = a 2 + b 2, so c 2 = = = 5 and, the foci are unit below and above the center: The asymptotes are: Sketch the asymptotes, plot the vertices, draw the graph.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued)

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Applications Some comets travel in hyperbolic paths with the sun at one focus. Such comets pass by the sun only one time, unlike those with elliptical orbits, which reappear at intervals. A cross section of an amphitheater might be one branch of a hyperbola. A cross section of a nuclear cooling tower might also be a hyperbola.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Applications Another application of hyperbolas is in the long-range navigation system LORAN. This system uses transmitting stations in three locations to send out simultaneous signals to a ship or aircraft. The difference in the arrival times of the signals from one pair of transmitters is recorded on the ship or aircraft. This difference is also recorded for signals from another pair of transmitters. For each pair, a computation is performed to determine the difference in the distances from each member of the pair to the ship or aircraft. If each pair of differences is kept constant, two hyperbolas can be drawn. Each has one of the pairs of transmitters as foci,and the ship or aircraft lies on the intersection of two of their branches. See diagram on next slide.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Applications