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Copyright © 2011 Pearson Education, Inc. Slide 6.2-2 Chapter 6: Analytic Geometry 6.1Circles and Parabolas 6.2Ellipses and Hyperbolas 6.3Summary of the Conic Sections 6.4Parametric Equations

Copyright © 2011 Pearson Education, Inc. Slide 6.2-3 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-4 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-5 6.2 The Equation of an Ellipse

Copyright © 2011 Pearson Education, Inc. Slide 6.2-6 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-7 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-8 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-9 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-10 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-11 6.2 Ellipse Centered at (h, k) An ellipse centered at (h, k) and either a horizontal or vertical major axis satisfies one of the following equations, where a > b > 0, and with c > 0:

Copyright © 2011 Pearson Education, Inc. Slide 6.2-12 6.2 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).

Copyright © 2011 Pearson Education, Inc. Slide 6.2-13 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-14 6.2 Finding the Standard Form of an Ellipse ExampleWrite the equation in standard form. Solution Center (2, –3); Vertices (2±3, –3) = (5, –3), (–1, – 3)

Copyright © 2011 Pearson Education, Inc. Slide 6.2-15 6.2 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 such that the absolute value of the difference of the distances from two fixed points is constant. The two fixed points are called the foci of the hyperbola.

Copyright © 2011 Pearson Education, Inc. Slide 6.2-16 6.2 Standard Forms of Equations for Hyperbolas The hyperbola with center at the origin and equation has vertices (±a, 0), asymptotes y = ±b/ax, and foci (±c, 0), where c 2 = a 2 + b 2. The hyperbola with center at the origin and equation has vertices (0, ±a), asymptotes y = ±a/bx, and foci (0, ±c), where c 2 = a 2 + b 2.

Copyright © 2011 Pearson Education, Inc. Slide 6.2-17 6.2 Standard Forms 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

Copyright © 2011 Pearson Education, Inc. Slide 6.2-18 6.2 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 © 2011 Pearson Education, Inc. Slide 6.2-19 6.2 Graphing a Hyperbola with the Graphing Calculator ExampleGraph SolutionSolve the given equation for y.

Copyright © 2011 Pearson Education, Inc. Slide 6.2-20 6.2 Graphing a Hyperbola Translated from the Origin ExampleGraph SolutionThis hyperbola has the same graph as except that it is centered at (–3, –2).

Copyright © 2011 Pearson Education, Inc. Slide 6.2-21 6.2 Finding the Standard Form for a Hyperbola ExampleWrite the equation in standard form. Solution Center (1, –2); Vertices (1±2, –2) = (3, –2), (–1, – 2)