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Hyperbolas 753 333202_1004.qxd 12/8/05 9:03 AM Page 753 Section 10.4 Hyperbolas 753 10.4 Hyperbolas What you should learn •Write equations of hyperbolas Introduction in standard form. The third type of conic is called a hyperbola. The definition of a hyperbola is •Find asymptotes of and graph similar to that of an ellipse. The difference is that for an ellipse the sum of the hyperbolas. distances between the foci and a point on the ellipse is fixed, whereas for a •Use properties of hyperbolas hyperbola the difference of the distances between the foci and a point on the to solve real-life problems. hyperbola is fixed. •Classify conics from their general equations. Definition of Hyperbola Why you should learn it A hyperbola is the set of all points ͑x, y͒ in a plane, the difference of whose Hyperbolas can be used to distances from two distinct fixed points (foci) is a positive constant. See model and solve many types of Figure 10.29. real-life problems. For instance, in Exercise 42 on page 761, hyperbolas are used in long c distance radio navigation for (,x y ) Brancha Branch d aircraft and ships. 2 d1 Vertex Focus Focus Center Vertex Transverse axis − dd21is a positive constant. FIGURE 10.29 FIGURE 10.30 The graph of a hyperbola has two disconnected branches. The line through the two foci intersects the hyperbola at its two vertices. The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola. See Figure 10.30. The development of the standard form of the equation of a hyperbola is similar to that of an ellipse. Note in the definition below that a, b, and c are related differently for hyperbolas than AP/Wide World Photos for ellipses. Standard Equation of a Hyperbola The standard form of the equation of a hyperbola with center ͑h, k͒ is ͑x Ϫ h͒2 ͑ y Ϫ k͒2 Ϫ ϭ 1 Transverse axis is horizontal. a 2 b2 ͑ y Ϫ k͒ 2 ͑x Ϫ h͒2 Ϫ ϭ 1. Transverse axis is vertical. a2 b 2 The vertices are a units from the center, and the foci are c units from the center. Moreover,c 2 ϭ a2 ϩ b2. If the center of the hyperbola is at the origin ͑0, 0͒, the equation takes one of the following forms. x 2 y 2 Transverse axis y 2 x 2 Transverse axis Ϫ ϭ 1 Ϫ ϭ 1 is vertical. a2 b2 is horizontal. a2 b2 333202_1004.qxd 12/8/05 9:03 AM Page 754 754 Chapter 10 Topics in Analytic Geometry Figure 10.31 shows both the horizontal and vertical orientations for a hyperbola. ()xh− 22()yk− ()yk− 22()xh− − = 1 − = 1 ab22 ab22 y y (,h k + c ) (,)hc− k (,h k ) (+,)hc k (,h k ) x x (,h k− c ) Transverse axis is horizontal. Transverse axis is vertical. FIGURE 10.31 Example 1 Finding the Standard Equation of a Hyperbola Find the standard form of the equation of the hyperbola with foci ͑Ϫ1, 2͒ and ͑5, 2͒ and vertices ͑0, 2͒ and ͑4, 2͒. When finding the standard form Solution of the equation of any conic, By the Midpoint Formula, the center of the hyperbola occurs at the point ͑2, 2͒. it is helpful to sketch a graph Furthermore,c ϭ 5 Ϫ 2 ϭ 3 and a ϭ 4 Ϫ 2 ϭ 2, and it follows that of the conic with the given b ϭ Ίc2 Ϫ a2 ϭ Ί 2 Ϫ 2 ϭ Ί Ϫ ϭ Ί characteristics. 3 2 9 4 5. So, the hyperbola has a horizontal transverse axis and the standard form of the equation is ͑x Ϫ 2͒2 ͑y Ϫ 2͒2 Ϫ ϭ 1. See Figure 10.32. 22 ͑Ί5͒2 This equation simplifies to ͑x Ϫ 2͒2 ͑y Ϫ 2͒2 Ϫ ϭ 1. 4 5 (x − 2)2 (y − 2)2 − = 1 y 22 ( 5 ( 2 5 4 (0, 2) (4, 2) (2, 2) (−1, 2) (5, 2) x 123 4 −1 FIGURE 10.32 Now try Exercise 27. 333202_1004.qxd 12/8/05 9:03 AM Page 755 Section 10.4 Hyperbolas 755 Conjugate Asymptotes of a Hyperbola axis Each hyperbola has two asymptotes that intersect at the center of the hyperbola, (h, k + b) Asymptote as shown in Figure 10.33. The asymptotes pass through the vertices of a rectangle of dimensions 2a by 2b, with its center at ͑h, k͒. The line segment of length 2b joining ͑h, k ϩ b͒ and ͑h, k Ϫ b͒ ͓or ͑h ϩ b, k͒ and ͑h Ϫ b, k͔͒ is the conjugate (h, k) axis of the hyperbola. Asymptotes of a Hyperbola (h, k − b) Asymptote The equations of the asymptotes of a hyperbola are − + b (h a, k) (h a, k) y ϭ k ± ͑x Ϫ h͒ Transverse axis is horizontal. a FIGURE 10.33 a y ϭ k ± ͑x Ϫ h͒. Transverse axis is vertical. b Example 2 Using Asymptotes to Sketch a Hyperbola Sketch the hyperbola whose equation is 4x 2 Ϫ y 2 ϭ 16. Solution Divide each side of the original equation by 16, and rewrite the equation in standard form. x 2 y 2 Ϫ ϭ 1 Write in standard form. 22 42 From this, you can conclude that a ϭ 2, b ϭ 4, and the transverse axis is hori- zontal. So, the vertices occur at ͑Ϫ2, 0͒ and ͑2, 0͒, and the endpoints of the con- jugate axis occur at ͑0, Ϫ4͒ and ͑0, 4͒. Using these four points, you are able to sketch the rectangle shown in Figure 10.34. Now, from c2 ϭ a2 ϩ b2, you have c ϭ Ί22 ϩ 42 ϭ Ί20 ϭ 2Ί5. So, the foci of the hyperbola are ͑Ϫ2Ί5, 0͒ and ͑2Ί5, 0͒. Finally, by drawing the asymptotes through the corners of this rec- tangle, you can complete the sketch shown in Figure 10.35. Note that the asymp- totes are y ϭ 2x and y ϭϪ2x. y y 8 8 (0, 4) 6 6 (−2, 0) (2, 0) (− 2 5, 0 ) (2 5, 0 ) x x −6 −4 46 −6 −4 46 xy22 − = 1 22 42 (0,− 4) −6 −6 FIGURE 10.34 FIGURE 10.35 Now try Exercise 7. 333202_1004.qxd 12/8/05 11:22 AM Page 756 756 Chapter 10 Topics in Analytic Geometry Example 3 Finding the Asymptotes of a Hyperbola Sketch the hyperbola given by 4x 2 Ϫ 3y 2 ϩ 8x ϩ 16 ϭ 0 and find the equations of its asymptotes and the foci. Solution 4 x 2 Ϫ 3y 2 ϩ 8x ϩ 16 ϭ 0 Write original equation. ͑4x2 ϩ 8x͒ Ϫ 3y2 ϭϪ16 Group terms. 4 ͑x 2 ϩ 2x͒ Ϫ 3y 2 ϭϪ16 Factor 4 from x-terms. 4͑x 2 ϩ 2x ϩ 1͒ Ϫ 3y 2 ϭϪ16 ϩ 4 Add 4 to each side. 4 ͑x ϩ 1͒2 Ϫ 3y 2 ϭϪ12 Write in completed square form. ͑x ϩ 1͒2 y2 Ϫ ϩ ϭ 1 Divide each side by Ϫ12. 3 4 y 2 ͑x ϩ 1͒2 y Ϫ ϭ 1 Write in standard form. 22 ͑Ί3͒2 (− 1, 2 + 7 ) From this equation you can conclude that the hyperbola has a vertical transverse (−1, 2) 4 ͑Ϫ ͒ ͑Ϫ ͒ ͑Ϫ Ϫ ͒ 3 axis, centered at 1, 0 , has vertices 1, 2 and 1, 2 , and has a conjugate y2 (x + 1)2 axis with endpoints ͑Ϫ1 Ϫ Ί3, 0͒ and ͑Ϫ1 ϩ Ί3, 0͒. To sketch the hyperbola, − = 1 (−1, 0) 1 22 ( 3 )2 draw a rectangle through these four points. The asymptotes are the lines passing x through the corners of the rectangle. Using a ϭ 2 and b ϭ Ί3, you can −4 −3 −2 12345 conclude that the equations of the asymptotes are 2 2 − y ϭ ͑x ϩ 1͒ and y ϭ Ϫ ͑x ϩ 1͒. 3 Ί Ί (−1, −2) 3 3 Finally, you can determine the foci by using the equation c2 ϭ a2 ϩ b2. So, you (− 1, − 2 − 7 ) have c ϭ Ί22 ϩ ͑Ί3͒2 ϭ Ί7, and the foci are ͑Ϫ1, Ϫ2 Ϫ Ί7͒ and FIGURE 10.36 ͑Ϫ1, Ϫ2 ϩ Ί7͒. The hyperbola is shown in Figure 10.36. Now try Exercise 13. Technology You can use a graphing utility to graph a hyperbola by graphing the upper and lower portions in the same viewing window. For instance, to graph the hyperbola in Example 3, first solve for y to get ͑ ϩ ͒2 ͑x ϩ 1͒2 ϭ ϩ x 1 ϭϪ ϩ y 2Ί1 and y2 2Ί1 . 1 3 3 Use a viewing window in which Ϫ9 ≤ x ≤ 9 and Ϫ6 ≤ y ≤ 6. You should obtain the graph shown below. Notice that the graphing utility does not draw the asymptotes. However, if you trace along the branches, you will see that the values of the hyperbola approach the asymptotes. 6 −99 −6 333202_1004.qxd 12/8/05 9:03 AM Page 757 Section 10.4 Hyperbolas 757 y y = 2x − 8 Example 4 Using Asymptotes to Find the Standard Equation Find the standard form of the equation of the hyperbola having vertices ͑3, Ϫ5͒ 2 (3, 1) and ͑3, 1͒ and having asymptotes x y ϭ 2x Ϫ 8 and y ϭϪ2x ϩ 4 −2 264 as shown in Figure 10.37. −2 Solution −4 By the Midpoint Formula, the center of the hyperbola is ͑3, Ϫ2͒. Furthermore, the (3, −5) hyperbola has a vertical transverse axis with a ϭ 3.
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