SECTION 10.5 Rotation of Axes; General Form of a Conic 721 Proof • If = AC 0, then either = A 0 or = C 0, but not both, so the form of equation (2) is either + + + = ≠ Ax Dx Ey F A 0 0 2 or + + + = ≠ Cy Dx Ey F C 0 0 2 Using the results of Problems 82 and 83 at the end of Section 10.2, it follows that, except for the degenerate cases, the equation is a parabola. • If > AC 0, then A and C have the same sign. Using the results of Problem 92 at the end of Section 10.3, except for the degenerate cases, the equation is an ellipse. • If < AC 0, then A and C have opposite signs. Using the results of Problem 90 at the end of Section 10.4, except for the degenerate cases, the equation is a hyperbola. ■ THEOREM Identifying Conics without Completing the Squares Excluding degenerate cases, the equation + + + + = Ax Cy Dx Ey F 0 2 2 where A and C are not both equal to zero: • Defines a parabola if = AC 0. • Defines an ellipse (or a circle) if > AC 0. • Defines a hyperbola if < AC 0. Although we are not studying the degenerate cases of equation (2), in practice, you should be alert to the possibility of degeneracy. Identifying a Conic without Completing the Squares Identify the graph of each equation without completing the squares. (a) + + − = x y x y 3 6 6 12 0 2 2 (b) − + + = x y y 2 3 6 4 0 2 2 (c) − + = y x2 4 0 2 Solution EXAMPLE 1 (a) Note that = A 3 and = C 6. Since = > AC 18 0, the equation defines an ellipse. (b) Here = A 2 and = − C 3, so = − < AC 6 0. The equation defines a hyperbola. (c) Here = A 0 and = C 1, so = AC 0. The equation defines a parabola. Now Work PROBLEM 11 Although we can now identify the type of conic represented by any general second-degree equation of the form of equation (2) without completing the squares, we still need to complete the squares if we desire additional information about the conic, such as its graph. 2 Use a Rotation of Axes to Transform Equations Now let’s suppose that ≠ B 0. To discuss this case, we introduce a new procedure: rotation of axes . In a rotation of axes , the origin remains fixed while the x -axis and y -axis are rotated through an angle θ to a new position; the new positions of the x -axis and the y -axis are denoted by ′x and ′y, respectively, as shown in Figure 51(a). Figure 51 (a) y x u u O y9 x9
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