SECTION 10.4 The Hyperbola 705 Figure 34 illustrates a hyperbola with foci F1 and F .2 The line containing the foci is called the transverse axis . The midpoint of the line segment joining the foci is the center of the hyperbola. The line through the center and perpendicular to the transverse axis is the conjugate axis . The hyperbola consists of two separate curves, called branches , that are symmetric with respect to the transverse axis, conjugate axis, and center. The two points of intersection of the hyperbola and the transverse axis are the vertices , V1 and V ,2 of the hyperbola. 1 Analyze Hyperbolas with Center at the Origin With these ideas in mind, we are now ready to find the equation of a hyperbola in the rectangular coordinate system. First, place the center at the origin. Next, position the hyperbola so that its transverse axis coincides with a coordinate axis. Suppose that the transverse axis coincides with the x -axis, as shown in Figure 35. If c is the distance from the center to a focus, one focus will be at F c, 0 1 ( ) = − and the other at F c, 0 . 2 ( ) = Now let the constant difference of the distances from any point P x y , ( ) = on the hyperbola to the foci F1 and F2 be denoted by a2 , ± where a 0. > (If P is on the right branch, the + sign is used; if P is on the left branch, the − sign is used.) The coordinates of P must satisfy the equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − = ± + + − − + = ± + + = ± + − + + + = ± − + + − + + + + = ± − + + − + + − = ± − + − = ± − + d F P d F P a x c y x c y a x c y a x c y x c y a a x c y x c y x cx c y a a x c y x cx c y cx a a x c y cx a a x c y , , 2 2 2 4 4 2 4 4 2 4 4 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 OBJECTIVES 1 Analyze Hyperbolas with Center at the Origin (p. 705) 2 Find the Asymptotes of a Hyperbola (p. 710) 3 Analyze Hyperbolas with Center at ( ) h k , (p. 712) 4 Solve Applied Problems Involving Hyperbolas (p. 714) 10.4 The Hyperbola Now Work the ‘Are You Prepared?’ problems on page 716. • Distance Formula (Section 1.2, p. 14) • Completing the Square ( Section A.3, p. A29) • Intercepts ( Section 1.3 , pp. 20 – 21 ) • Symmetry (Section 1.3, pp. 21–23) • Asymptotes (Section 4.5, pp. 238–241) • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) • Square Root Method (Section A.6, p. A49) PREPARING FOR THIS SECTION Before getting started, review the following: DEFINITION A hyperbola is the collection of all points in a plane the difference of whose distances from two fixed points, called the foci , is a constant. Difference of the distances from P to the foci equals ± a2 . Use the Distance Formula. Isolate one radical. Square both sides. Multiply. Simplify; isolate the radical. Divide both sides by 4. Figure 34 Hyperbola F2 Transverse axis Conjugate axis F1 Center V1 V2 Figure 35 ( ) ( ) − =± d F P d F P a , , 2 1 2 x y P 5 (x, y) Transverse axis d(F2, P) d(F1, P) F2 5 (c, 0) F1 5 (2c, 0) (continued)
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