Copyright © 2007 Pearson Education, Inc. Slide 6-1

Copyright © 2007 Pearson Education, Inc. Slide 6-2 Chapter 6: Analytic Geometry 6.1Circles and Parabolas 6.2Ellipses and Hyperbolas 6.3Summary of the Conic Sections 6.4Parametric Equations

Copyright © 2007 Pearson Education, Inc. Slide Ellipses and Hyperbolas The graph of an ellipse is not that of a function. The foci lie on the major axis – the line from V to V. The minor axis – B to B An ellipse is the set of all points in a plane the sum of whose distances from two fixed points is constant. Each fixed point is called a focus of the ellipse.

Copyright © 2007 Pearson Education, Inc. Slide The Equation of an Ellipse Let the foci of an ellipse be at the points (  c, 0). The sum of the distances from the foci to a point (x, y) on the ellipse is 2a. So, we rewrite the following equation.

Copyright © 2007 Pearson Education, Inc. Slide The Equation of an Ellipse

Copyright © 2007 Pearson Education, Inc. Slide The Equation of an Ellipse Replacing a 2 – c 2 with b 2 gives the standard equation of an ellipse with the foci on the x-axis. Similarly, if the foci were on the y-axis, we get

Copyright © 2007 Pearson Education, Inc. Slide The Equation of an Ellipse The ellipse with center at the origin and equation has vertices (  a, 0), endpoints of the minor axis (0,  b), and foci (  c, 0), where c 2 = a 2 – b 2. The ellipse with center at the origin and equation has vertices (0,  a), endpoints of the minor axis (  b, 0), and foci (0,  c), where c 2 = a 2 – b 2.

Copyright © 2007 Pearson Education, Inc. Slide Graphing an Ellipse Centered at the Origin ExampleGraph SolutionDivide both sides by 36. This ellipse, centered at the origin, has x-intercepts 3 and –3, and y-intercepts 2 and –2. The domain is [–3, 3]. The range is [–2, 2].

Copyright © 2007 Pearson Education, Inc. Slide Finding Foci of an Ellipse ExampleFind the coordinates of the foci of the equation SolutionFrom the previous example, the equation of the ellipse in standard form is Since 9 > 4, a 2 = 9 and b 2 = 4. The major axis is along the x-axis, so the foci have coordinates

Copyright © 2007 Pearson Education, Inc. Slide Finding the Equation of an Ellipse ExampleFind the equation of the ellipse having center at the origin, foci at (0, ±3), and major axis of length 8 units. Give the domain and range. Solution2a = 8, so a = 4. Foci lie on the y-axis, so the larger intercept, a, is used to find the denominator for y 2. The standard form is with domain and range [–4, 4].

Copyright © 2007 Pearson Education, Inc. Slide Ellipse Centered at (h, k) ExampleGraph Analytic SolutionCenter at (2, –1). Since a > b, a = 4 is associated with the y 2 term, so the vertices are on the vertical line through (2, –1). An ellipse centered at (h, k) with horizontal major axis of length 2a and vertical minor axis of length 2b has equation Similarly for ellipses with vertical major axis.

Copyright © 2007 Pearson Education, Inc. Slide Ellipse Centered at (h, k) Graphical Solution Solving for y in the equation yields The + sign indicates the upper half of the ellipse, while the – sign yields the bottom half.

Copyright © 2007 Pearson Education, Inc. Slide Hyperbolas If the center is at the origin, the foci are at (±c, 0). The midpoint of the line segment F F is the center of the hyperbola. The vertices are at (±a, 0). The line segment VV is called the transverse axis. A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points is constant. The two fixed points are called the foci of the hyperbola.

Copyright © 2007 Pearson Education, Inc. Slide Standard Form of Equations for Hyperbolas Solving for y in the first equation gives If |x| is large, the difference approaches x 2. Thus, the hyperbola has asymptotes The hyperbola with center at the origin and equation has vertices (±a, 0) and foci (±c, 0), where c 2 = a 2 + b 2. The hyperbola with center at the origin and equation has vertices (0, ±a) and foci (0, ±c), where c 2 = a 2 + b 2.

Copyright © 2007 Pearson Education, Inc. Slide Using Asymptotes to Graph a Hyperbola ExampleSketch the asymptotes and graph the hyperbola Solution a = 5 and b = 7 Choosing x = 5 (or –5) gives y = ±7. These four points: (5, 7), (–5, 7), (5, –7), and (–5, –7), are the corners of the fundamental rectangle shown. The x-intercepts are ±5.

Copyright © 2007 Pearson Education, Inc. Slide Graphing a Hyperbola with the Graphing Calculator ExampleGraph SolutionSolve the given equation for y.

Copyright © 2007 Pearson Education, Inc. Slide Graphing a Hyperbola Translated from the Origin ExampleGraph SolutionThis hyperbola has the same graph as except that it is centered at (–3, –2).