1025 10.4 Summary of the Conic Sections Characteristics The graphs of parabolas, circles, ellipses, and hyperbolas are called conic sections because each graph can be obtained by intersecting a cone with a plane, as suggested by Figure 1 at the beginning of the chapter. All conic sections of the types presented in this chapter have equations of the general form Ax2 +Cy2 +Dx +Ey +F =0, where either A or C must be nonzero. Relating Concepts For individual or collaborative investigation (Exercises 69–74) The graph of x 2 4 - y 2 = 1 is a hyperbola. We know that the graph of this hyperbola approaches its asymptotes as 0 x 0 increases without bound. Work Exercises 69–74 in order, to see the relationship between the hyperbola and one of its asymptotes. 69. Solve x 2 4 - y 2 = 1 for y, and choose the positive square root. 70. Find the equation of the asymptote with positive slope. 71. Use a calculator to evaluate the y-coordinate of the point where x = 50 on the graph of the portion of the hyperbola represented by the equation obtained in Exercise 69. Round the answer to the nearest hundredth. 72. Find the y-coordinate of the point where x = 50 on the graph of the asymptote found in Exercise 70. 73. Compare the results in Exercises 71 and 72. How do they support the following statement? When x = 50, the graph of the function defined by the equation found in Exercise 69 lies below the graph of the asymptote found in Exercise 70. 74. What happens if we choose x-values greater than 50? 68. Suppose a hyperbola has center at the origin, foci at F′1-c, 02 and F1c, 02, and 0 d1P, F′2 - d1P, F20 = 2a. Let b2 = c2 - a2, and show that an equation of the hyperbola is x2 a2 - y2 b2 = 1. 10.4 Summary of the Conic Sections ■ Characteristics ■ Identifying Conic Sections ■ Geometric Definition of Conic Sections Summary of Special Characteristics of Conic Sections Conic Section Characteristic Example Parabola Either A = 0 or C = 0, but not both. x2 - y - 4 = 0 y2 - x - 4y = 0 Circle A = C≠0 x2 + y2 - 16 = 0 Ellipse A≠C, AC70 25x2 + 16y2 - 400 = 0 Hyperbola AC60 x2 - y2 - 1 = 0
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