CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates Section 10.1 Conics and Calculus . 360 Section 10.2 Plane Curves and Parametric Equations . 383 Section 10.3 Parametric Equations and Calculus . 392 Section 10.4 Polar Coordinates and Polar Graphs . 407 Section 10.5 Area and Arc Length in Polar Coordinates . 423 Section 10.6 Polar Equations of Conics and Kepler’s Laws . 436 Review Exercises . 444 Problem Solving . 461 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates Section 10.1 Conics and Calculus 1. y2 ϭ 4x 2. x2 ϭ 8y 3. ͑x ϩ 3͒2 ϭϪ2͑y Ϫ 2͒ Vertex: ͑0, 0͒ Vertex: ͑0, 0͒ Vertex: ͑Ϫ3, 2͒ ϭ ϭ ϭϪ1 p 1 > 0 p 2 > 0 p 2 < 0 Opens to the right Opens upward Opens downward Matches graph (h). Matches graph (a). Matches graph (e). ͑x Ϫ 2͒2 ͑y ϩ 1͒2 x2 y2 x2 y2 4. ϩ ϭ 1 5. ϩ ϭ 1 6. ϩ ϭ 1 16 4 9 4 9 9 Center: ͑2, Ϫ1͒ Center: ͑0, 0͒ Circle radius 3. Ellipse Ellipse Matches (g) Matches (b) Matches (f) y2 x2 ͑x Ϫ 2͒2 y2 7. Ϫ ϭ 1 8. Ϫ ϭ 1 16 1 9 4 Hyperbola Hyperbola Center: ͑0, 0͒ Center: ͑Ϫ2, 0͒ Vertical transverse axis Horizontal transverse axis Matches (c) Matches (d) 2 ϭϪ ϭ ͑Ϫ3͒ 2 ϩ ϭ 9. y 6x 4 2 x 10. x 8y 0 Vertex: ͑0, 0͒ y x2 ϭ 4͑Ϫ2͒y y 3 8 Focus: ͑Ϫ , 0͒ (0, 0) 2 Vertex: ͑0, 0͒ x 3 −8 −4 4 8 ϭ 4 Directrix: x 2 ͑ Ϫ ͒ Focus: 0, 2 − (0, 0) 4 x ϭ 12 8 4 Directrix: y 2 −8 4 −12 8 11. ͑x ϩ 3͒ ϩ ͑ y Ϫ 2͒2 ϭ 0 12. ͑x Ϫ 1͒2 ϩ 8͑y ϩ 2͒ ϭ 0 ͑ Ϫ ͒2 ϭ ͑Ϫ1͒͑ ϩ ͒ ͑ Ϫ ͒2 ϭ ͑Ϫ ͒͑ ϩ ͒ y 2 4 4 x 3 x 1 4 2 y 2 Vertex: ͑Ϫ3, 2͒ y Vertex: ͑1, Ϫ2͒ y Focus: ͑Ϫ3.25, 2͒ 4 Focus: ͑1, Ϫ4͒ x −8 −4 4 8 Directrix: x ϭϪ2.75 (−3, 2) 2 Directrix: y ϭ 0 −4 x (1,− 2) −8 −6 −4 −2 −8 −2 −12 −4 360 Section 10.1 Conics and Calculus 361 13. y2 Ϫ 4y Ϫ 4x ϭ 0 14. y2 ϩ 6y ϩ 8x ϩ 25 ϭ 0 y2 Ϫ 4y ϩ 4 ϭ 4x ϩ 4 y2 ϩ 6y ϩ 9 ϭϪ8x Ϫ 25 ϩ 9 ͑y Ϫ 2͒2 ϭ 4͑1͒͑x ϩ 1͒ ͑y ϩ 3͒2 ϭ 4͑Ϫ2͒͑x ϩ 2͒ Vertex: ͑Ϫ1, 2͒ y Vertex: ͑Ϫ2, Ϫ3͒ y Focus: ͑0, 2͒ 6 Focus: ͑Ϫ4, Ϫ3͒ 8 Directrix: x ϭϪ2 4 Directrix: x ϭ 0 4 (−1, 2) x −20 −16 −12 −8 −4 (2,3)−− x −8 2 2 4 6 −12 2 15. x2 ϩ 4x ϩ 4y Ϫ 4 ϭ 0 16. y2 ϩ 4y ϩ 8x Ϫ 12 ϭ 0 x2 ϩ 4x ϩ 4 ϭϪ4y ϩ 4 ϩ 4 y2 ϩ 4y ϩ 4 ϭϪ8x ϩ 12 ϩ 4 ͑x ϩ 2͒2 ϭ 4͑Ϫ1͒͑y Ϫ 2͒ ͑y ϩ 2͒2 ϭ 4͑Ϫ2͒͑x Ϫ 2͒ Vertex: ͑Ϫ2, 2͒ y Vertex: ͑2, Ϫ2͒ y 4 Focus: ͑Ϫ2, 1͒ 4 Focus: ͑0, Ϫ2͒ (−2, 2) 2 Directrix: y ϭ 3 Directrix: x ϭ 4 x −6 −4 −2 4 6 x (2,− 2) −6 −4 −2 2 −4 −2 −6 − 4 −8 2 ϩ ϩ ϭ ϭϪ1͑ 2 Ϫ ϩ ͒ ϭϪ1͑ 2 Ϫ ϩ Ϫ ͒ 17. y x y 0 18. y 6 x 8x 6 6 x 8x 16 10 2 ϩ ϩ 1 ϭϪ ϩ 1 2 y y 4 x 4 Ϫ6y ϭ ͑x Ϫ 4͒ Ϫ 10 2 ͑ ϩ 1͒ ϭ ͑Ϫ1͒͑ Ϫ 1͒ 2 y 2 4 4 x 4 Ϫ6y ϩ 10 ϭ ͑x Ϫ 4͒ 1 1 5 ͑ Ϫ ͒ 2 ͑ Ϫ ͒2 ϭϪ ͑ Ϫ ͒ Vertex: 4, 2 x 4 6 y 3 ͑ Ϫ1͒ Focus: 0, 2 ͑ Ϫ ͒2 ϭ ͑Ϫ3͒͑ Ϫ 5͒ − x 4 4 2 y 3 ϭ 1 5 2 Directrix: x 2 5 ͑ ͒ 4 Vertex: 4, 3 ϭϪ1 ± Ί1 Ϫ y1 2 4 x Focus: ͑4, 1͒ −3 6 1 1 19 ϭϪ Ϫ Ί Ϫ ϭ −2 10 y2 2 4 x Directrix: y 6 −4 19. y2 Ϫ 4x Ϫ 4 ϭ 0 20. x2 Ϫ 2x ϩ 8y ϩ 9 ϭ 0 2 2 y ϭ 4x ϩ 4 x Ϫ 2x ϩ 1 ϭϪ8y Ϫ 9 ϩ 1 2 ϭ 4͑1͒͑x ϩ 1͒ ͑x Ϫ 1͒ ϭ 4͑Ϫ2͒͑y ϩ 1͒ Vertex: ͑Ϫ1, 0͒ 4 Vertex: ͑1, Ϫ1͒ 2 Focus: ͑0, 0͒ Focus: ͑1, Ϫ3͒ −8 10 Directrix: x ϭϪ2 −6 6 Directrix: y ϭ 1 ϭ Ί ϩ y1 2 x 1 −4 −10 ϭϪ Ί ϩ y2 2 x 1 362 Chapter 10 Conics, Parametric Equations, and Polar Coordinates 21. ͑ y Ϫ 2͒2 ϭ 4͑Ϫ2͒͑x Ϫ 3͒ 22. ͑ x ϩ 1͒2 ϭ 4͑Ϫ2͒͑y Ϫ 2͒ y2 Ϫ 4y ϩ 8x Ϫ 20 ϭ 0 x2 ϩ 2x ϩ 8y Ϫ 15 ϭ 0 23. ͑x Ϫ h͒2 ϭ 4p͑y Ϫ k͒ 24. Vertex: ͑0, 2͒ x2 ϭ 4͑6͒͑y Ϫ 4͒ ͑ y Ϫ 2͒2 ϭ 4͑2͒͑x Ϫ 0͒ x2 Ϫ 24y ϩ 96 ϭ 0 y2 Ϫ 8x Ϫ 4y ϩ 4 ϭ 0 25. y ϭ 4 Ϫ x2 26. y ϭ 4 Ϫ ͑x Ϫ 2͒2 ϭ 4x Ϫ x2 x2 ϩ y Ϫ 4 ϭ 0 x2 Ϫ 4x ϩ y ϭ 0 27. Since the axis of the parabola is vertical, the form of the 28. From Example 2:4p ϭ 8 or p ϭ 2 equation is y ϭ ax2 ϩ bx ϩ c. Now, substituting the val- Vertex: ͑4, 0͒ ues of the given coordinates into this equation, we obtain ͑ x Ϫ 4͒2 ϭ 8͑y Ϫ 0͒ 3 ϭ c, 4 ϭ 9a ϩ 3b ϩ c, 11 ϭ 16a ϩ 4b ϩ c. 2 Ϫ Ϫ ϩ ϭ ϭ 5 ϭϪ14 ϭ x 8x 8y 16 0 Solving this system, we have a 3, b 3 , c 3. Therefore, ϭ 5 2 Ϫ 14 ϩ 2 Ϫ Ϫ ϩ ϭ y 3x 3 x 3 or 5x 14x 3y 9 0. 29. x2 ϩ 4y2 ϭ 4 y 30. 5x2 ϩ 7y2 ϭ 70 y 6 x2 y2 2 x2 y2 ϩ ϭ 1 ϩ ϭ 1 4 4 1 14 10 2 (0, 0) 2 ϭ 2 ϭ 2 ϭ x 2 ϭ 2 ϭ 2 ϭ x a 4, b 1, c 3 −11 a 14, b 10, c 4 −6 −2 2 4 6 Center: ͑0, 0͒ Center: ͑0, 0͒ −4 − Foci: ͑±Ί3, 0͒ 2 Foci: ͑±2, 0͒ −6 Vertices: ͑±2, 0͒ Vertices: ͑±Ί14, 0͒ Ί3 2 Ί14 e ϭ e ϭ ϭ 2 Ί14 7 ͑x Ϫ 1͒2 ͑y Ϫ 5͒2 y ͑x ϩ 2͒2 ͑y ϩ 4͒2 y 31. ϩ ϭ 1 32. ϩ ϭ 1 9 25 12 1 1͞4 x (1, 5) −5 −4 −3 −2 −1 − 2 ϭ 2 ϭ 2 ϭ 1 a 25, b 9, c 16 2 2 1 2 3 a ϭ 1, b ϭ , c ϭ −2 ͑ ͒ 4 4 4 (2,4)−− Center: 1, 5 −3 ͑Ϫ Ϫ ͒ ͑ ͒ ͑ ͒ x Center: 2, 4 −4 Foci: 1, 9 , 1, 1 8 4 4 8 Ί ͑ ͒ ͑ ͒ 3 −5 Vertices: 1, 10 , 1, 0 Foci: Ϫ2 ± , Ϫ4 ΂ 2 ΃ 4 e ϭ Vertices: ͑Ϫ1, Ϫ4͒, ͑Ϫ3, Ϫ4͒ 5 Ί3 e ϭ 2 Section 10.1 Conics and Calculus 363 33. 9x2 ϩ 4y2 ϩ 36x Ϫ 24y ϩ 36 ϭ 0 9͑x2 ϩ 4x ϩ 4͒ ϩ 4͑y2 Ϫ 6y ϩ 9͒ ϭϪ36 ϩ 36 ϩ 36 ϭ 36 ͑x ϩ 2͒2 ͑y Ϫ 3͒2 ϩ ϭ 1 4 9 a2 ϭ 9, b2 ϭ 4, c2 ϭ 5 y Center: ͑Ϫ2, 3͒ Foci: ͑Ϫ2, 3 ± Ί5 ͒ 6 ͑Ϫ ͒ ͑Ϫ ͒ Vertices: 2, 6 , 2, 0 (−2, 3) Ί5 2 e ϭ 3 x 6 4 2 2 34. 16x2 ϩ 25y2 Ϫ 64x ϩ 150y ϩ 279 ϭ 0 35. 12x2 ϩ 20y2 Ϫ 12x ϩ 40y Ϫ 37 ϭ 0 16͑x2 Ϫ 4x ϩ 4͒ ϩ 25͑y2 ϩ 6y ϩ 0͒ ϭϪ279 ϩ 64 ϩ 225 1 12΂x2 Ϫ x ϩ ΃ ϩ 20͑y2 ϩ 2y ϩ 1͒ ϭ 37 ϩ 3 ϩ 20 4 ϭ 10 ϭ 60 ͑x Ϫ 2͒2 ͑y ϩ 3͒2 ϩ ϭ 1 ͑5͞8͒ ͑2͞5͒ ͓x Ϫ ͑1͞2͔͒2 ͑y ϩ 1͒2 ϩ ϭ 1 5 3 5 2 9 a2, , b2 ϭ , c2 ϭ a2 Ϫ b2 ϭ 8 5 40 a2 ϭ 5, b2 ϭ 3, c2 ϭ 2 Center: ͑2, Ϫ3͒ 1 Center: ΂ , Ϫ1΃ 3Ί10 2 Foci: 2 ± , Ϫ3 ΂ 20 ΃ 1 Foci: ΂ ± Ί2, Ϫ1΃ Ί10 2 Vertices: 2 ± , Ϫ3 ΂ 4 ΃ 1 Vertices: ΂ ± Ί5, Ϫ1΃ 2 c 3 e ϭ ϭ a 5 Solve for y: y 20͑y2 ϩ 2y ϩ 1͒ ϭϪ12x2 ϩ 12x ϩ 37 ϩ 20 1 57 ϩ 12x Ϫ 12x2 ͑ ϩ ͒2 ϭ x y 1 −1 1 234 20 −1 57 ϩ 12x Ϫ 12x2 −2 (2, −3) y ϭϪ1 ± Ί 20 −3 −4 (Graph each of these separately.) 1 −33 −3 364 Chapter 10 Conics, Parametric Equations, and Polar Coordinates 36. 36x2 ϩ 9y2 ϩ 48x Ϫ 36y ϩ 43 ϭ 0 37. x2 ϩ 2y2 Ϫ 3x ϩ 4y ϩ 0.25 ϭ 0 4 4 9 1 9 36΂x2 ϩ x ϩ ΃ ϩ 9͑y2 Ϫ 4y ϩ 4͒ ϭϪ43 ϩ 16 ϩ 36 ΂x2 Ϫ 3x ϩ ΃ ϩ 2͑y2 ϩ 2y ϩ 1͒ ϭϪ ϩ ϩ 2 ϭ 4 3 9 4 4 4 ϭ 9 ͓x Ϫ ͑3͞2͔͒2 ͑y ϩ 1͒2 ϩ ϭ 1 4 2 ͓x ϩ ͑2͞3͔͒2 ͑y Ϫ 2͒2 ϩ ϭ 1 1͞4 1 a2 ϭ 4, b2 ϭ 2, c2 ϭ 2 1 3 3 a2 ϭ 1, b2 ϭ , c2 ϭ Center: ΂ , Ϫ1΃ 4 4 2 2 3 Ί Ϫ Center: ΂Ϫ , 2΃ Foci: ΂ ± 2, 1΃ 3 2 2 Ί3 Ϫ1 Ϫ 7 Ϫ Foci: Ϫ , 2 ± Vertices: ΂ , 1΃, ΂ , 1΃ ΂ 3 2 ΃ 2 2 2 2 1 Vertices: ΂Ϫ , 3΃, ΂Ϫ , 1΃ Solve for y: 2͑y2 ϩ 2y ϩ 1͒ ϭϪx2 ϩ 3x Ϫ ϩ 2 3 3 4 Solve for y: 1 7 ͑y ϩ 1͒2 ϭ ΂ ϩ 3x Ϫ x2΃ 2 4 9͑y2 Ϫ 4y ϩ 4͒ ϭϪ36x2 Ϫ 48x Ϫ 43 ϩ 36 7 ϩ 12x Ϫ 4x2 Ϫ͑36x2 ϩ 48x ϩ 7͒ y ϭϪ1 ± Ί ͑y Ϫ 2͒2 ϭ 8 9 (Graph each of these separately.) 1 y ϭ 2 ± ΊϪ͑36x2 ϩ 48x ϩ 7͒ 3 1 (Graph each of these separately.) −2 4 3 −3 −4 2 −1 38.