750 CHAPTER 10 Analytic Geometry Objectives Section You should be able to . . . Example(s) Review Exercises 10.1 1 Know the names of the conics (p. 681) 1–16 10.2 1 Analyze parabolas with vertex at the origin (p. 682) 1–6 1, 11 2 Analyze parabolas with vertex at h k , ( ) (p. 686) 7–9 4, 6, 9, 14 3 Solve applied problems involving parabolas (p. 688) 10 39 10.3 1 Analyze ellipses with center at the origin (p. 693) 1–5 3, 13 2 Analyze ellipses with center at h k , ( ) (p. 697) 6–8 8, 10, 16, 38 3 Solve applied problems involving ellipses (p. 700) 9 40 10.4 1 Analyze hyperbolas with center at the origin (p. 705) 1–5 2, 5, 12, 37 2 Find the asymptotes of a hyperbola (p. 710) 6, 7 2, 5, 7 3 Analyze hyperbolas with center at h k , ( ) (p. 712) 8, 9 7, 15, 17, 18 4 Solve applied problems involving hyperbolas (p. 714) 10 41 10.5 1 Identify a conic (p. 720) 1 19, 20 2 Use a rotation of axes to transform equations (p. 721) 2 24–26 3 Analyze an equation using a rotation of axes (p. 724) 3, 4 24–26, 44 4 Identify conics without rotating the axes (p. 726) 5 21–23 10.6 1 Analyze and graph polar equations of conics (p. 728) 1–3 27–29 2 Convert the polar equation of a conic to a rectangular equation (p. 733) 4 30, 31 10.7 1 Graph parametric equations by hand (p. 736) 1 32–34 2 Graph parametric equations using a graphing utility (p. 737) 2 32–34 3 Find a rectangular equation for a plane curve defined parametrically (p. 738) 3, 4 32–34 4 Use time as a parameter in parametric equations (p. 740) 5, 6 42, 43 5 Find parametric equations for curves defined by rectangular equations (p. 743) 7, 8 35, 36 Review Exercises In Problems 1–10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci; if it is a hyperbola, give its center, vertices, foci, and asymptotes. 1. y x 16 2 = − 2. x y 25 1 2 2 − = 3. y x 25 16 1 2 2 + = 4. x y4 4 2 + = 5. x y 4 8 2 2 − = 6. x x y 4 2 2 − = 7. y y x x 4 4 8 4 2 2 − − + = 8. x y x y 4 9 16 18 11 2 2 + − − = 9. x x y 4 16 16 32 0 2 − + + = 10. x y x y 9 4 18 8 23 2 2 + − + = 11. Parabola; focus at 2, 0 ; ( ) − directrix the line x 2 = 12. Hyperbola; center at 0, 0 ; ( ) focus at 0, 4 ; ( ) vertex at 0, 2 ( ) − 13. Ellipse; foci at 3, 0 ( ) − and 3,0 ; ( ) vertex at 4, 0 ( ) 14. Parabola; vertex at 2, 3 ; ( ) − focus at 2, 4 ( ) − In Problems 11–18, find an equation of the conic described. Graph the equation. 15. Hyperbola; center at 2, 3; ( ) − − focus at 4, 3; ( ) − − vertex at 3, 3 ( ) − − 16. Ellipse; foci at 4, 2 ( ) − and 4, 8 ; ( ) − vertex at 4, 10 ( ) − 17. Hyperbola; center at a c 1, 2 ; 3; 4; ( ) − = = transverse axis parallel to the x-axis 18. Hyperbola; vertices at 0, 1 ( ) and 6,1 ; ( ) asymptote the line y x 3 2 9 + = In Problems 19–23, identify each conic without completing the squares and without applying a rotation of axes. 19. y x y 4 3 8 0 2 + + − = 20. x y x y 2 4 8 2 0 2 2 + + − + = 21. x xy y x y 9 12 4 8 12 0 2 2 − + + + = 22. x xy y 4 10 4 9 0 2 2 + + − = 23. x xy y x y 2 3 2 4 1 0 2 2 − + + + − =
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