# 10 - CHAPTER 10 Conics, Parametric Equations, and Polar...

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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

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CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates Section 10.1 Conics and Calculus 360 10. Vertex: Focus: Directrix: y 5 2 s 0, 2 2 d s 0, 0 d x 4 8 4 8 4 8 12 (0, 0) y x 2 5 4 s 2 2 d y x 2 1 8 y 5 0 12. Focus: Directrix: y 5 0 s 1, 2 4 d x 4 8 4 8 4 8 12 (1, 2) y s 1, 2 2 d s x 2 1 d 2 5 4 s 2 2 ds y 1 2 d s x 2 1 d 2 1 8 s y 1 2 d 5 0 1. Opens to the right Matches graph (h). p 5 1 > 0 s 0, 0 d y 2 5 4 x 2. Opens upward Matches graph (a). p 5 2 > 0 s 0, 0 d x 2 5 8 y 3. Opens downward Matches graph (e). p 52 1 2 < 0 s 2 3, 2 d s x 1 3 d 2 2 s y 2 2 d 4. Center: Ellipse Matches (b) s 2, 2 1 d s x 2 2 d 2 16 1 s y 1 1 d 2 4 5 1 5. Center: Ellipse Matches (f) s 0, 0 d x 2 9 1 y 2 4 5 1 6. Circle radius 3. Matches (g) x 2 9 1 y 2 9 5 1 7. Hyperbola Center: Vertical transverse axis Matches (c) s 0, 0 d y 2 16 2 x 2 1 5 1 8. Hyperbola Center: Horizontal transverse axis Matches (d) s 2 2, 0 d s x 2 2 d 2 9 2 y 2 4 5 1 9. Focus: Directrix: x 5 3 2 s 2 3 2 , 0 d 8 4 8 4 4 8 12 x (0, 0) y s 0, 0 d y 2 6 x 5 4 s 2 3 2 d x 11. Focus: Directrix: x 2.75 s 2 3.25, 2 d x 2 4 2 2 4 6 8 4 ( 3, 2) y s 2 3, 2 d s y 2 2 d 2 5 4 s 2 1 4 d s x 1 3 d s x 1 3 d 1 s y 2 2 d 2 5 0
Section 10.1 Conics and Calculus 361 14. Vertex: Focus: Directrix: x 5 0 s 2 4, 2 3 d x 4 8 4 8 12 16 20 8 12 (2 , 3 ) −− y s 2 2, 2 3 d s y 1 3 d 2 5 4 s 2 2 ds x 1 2 d y 2 1 6 y 1 9 52 8 x 2 25 1 9 y 2 1 6 y 1 8 x 1 25 5 0 16. Focus: Directrix: x 5 4 s 0, 2 2 d x 4 4 2 6 4 6 2 8 6 4 (2, 2) y s 2, 2 2 d s y 1 2 d 2 5 4 s 2 2 ds x 2 2 d y 2 1 4 y 1 4 8 x 1 12 1 4 y 2 1 4 y 1 8 x 2 12 5 0 13. Focus: Directrix: x 2 s 0, 2 d 6 4 6 4 2 2 2 x ( 1, 2) y s 2 1, 2 d s y 2 2 d 2 5 4 s 1 ds x 1 1 d y 2 2 4 y 1 4 5 4 x 1 4 y 2 2 4 y 2 4 x 5 0 15. Focus: Directrix: y 5 3 s 2 2, 1 d x 2 4 2 4 ( 2, 2) 2 4 6 y s 2 2, 2 d s x 1 2 d 2 5 4 s 2 1 ds y 2 2 d x 2 1 4 x 1 4 4 y 1 4 1 4 x 2 1 4 x 1 4 y 2 4 5 0 17. Focus: Directrix: y 2 1 2 2 ! 1 4 2 x y 1 1 2 ± ! 1 4 2 x x 5 1 2 s 0, 2 1 2 d 5 2 3 2 s 1 4 , 2 1 2 d s y 1 1 2 d 2 5 4 s 2 1 4 ds x 2 1 4 d y 2 1 y 1 1 4 x 1 1 4 y 2 1 x 1 y 5 0 18. Focus: Directrix: y 5 19 6 s 4, 1 6 d 2 4 10 4 s 4, 5 3 d s x 2 4 d 2 5 4 s 2 3 2 ds y 2 5 3 d s x 2 4 d 2 6 s y 2 5 3 d 2 6 y 1 10 5 s x 2 4 d 2 2 6 y 5 s x 2 4 d 2 2 10 y 1 6 s x 2 2 8 x 1 6 d 1 6 s x 2 2 8 x 1 16 2 10 d 20. Focus: Directrix: y 5 1 s 1, 2 3 d 10 10 8 2 s 1, 2 1 d s x 2 1 d 2 5 4 s 2 2 ds y 1 1 d x 2 2 2 x 1 1 8 y 2 9 1 1 x 2 2 2 x 1 8 y 1 9 5 0 19.

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## This note was uploaded on 11/13/2010 for the course MATH MAT 231 taught by Professor Thurber during the Spring '08 term at Thomas Edison State.

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10 - CHAPTER 10 Conics, Parametric Equations, and Polar...

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