Math Conics Quiz Review

Conics Test REVIEW Name:_______________________________________ Hour:_______ Pre-Calculus Find the center and the exact radius of the circle. Then graph the circle. 25 1. (𝑥 − 5)2 + (𝑦 + 3)2 = 9 2. (𝑥 + 3)2 + 𝑦 2 = 9 Center:_________ Center:_________ Radius:_________ Radius:_________ Find the equation of the circle in standard form that satisfies the given conditions. 3. The circle has center (0, 0) and passes through (−3, 4). 4. The circle has center (−4, −3) and passes through (−1, −1). 5. The endpoints of the diameter of the circle are (2, 6) and (−8, 4). 6. The circle has center (4, −3) and and is tangent to the x-axis. Graph the ellipse and identify the center, vertices, and foci. 7. 𝑥2 9 𝑦2 + 25 = 1 8. 4𝑥 2 + 9𝑦 2 = 36 Center:___________ Center:___________ Vert:__________ Vert:__________ Foci:________ Foci:________ (𝑦−3)2 4 9. 𝑥 2 + 4𝑦 2 = 36 10. (𝑥 + 1)2 + Center:___________ Center:___________ Vert:__________ Vert:__________ Foci:________ Foci:________ 11. (𝑥−1)2 4 + (𝑦+2)2 9 =1 =1 12. 36(𝑥 + 4)2 + (𝑦 + 3)2 = 36 Center:___________ Center:___________ Vert:__________ Vert:__________ Foci:________ Foci:________ Find the standard form of the equation of each ellipse. 13. Foci (0, ±3), vertices (0, ±5) 14. Major axis horizontal with length 12; length of minor axis 4; center: (−1, 3) 15. Foci (±5, 0), length of major axis 12 16. Endpoints of major axis: (2, 2) & (8, 2) Endpoints of minor axis: (5, 3) & (5, 1) 17. 18. Graph the hyperbola and identify the center, vertices, slopes of asymptotes, and foci. 19. 𝑦2 9 𝑥2 20. 4𝑥 2 − 𝑦 2 = 16 − 16 = 1 Center: ___________ Center: ___________ Vertices:___________ Vertices:___________ Foci:__________ Foci:__________ Asymptotes:________ Asymptotes:________ 21. 4(𝑥 − 1)2 − 9(𝑦 + 2)2 = 36 22. Center:____________ Center: ____________ Vertices:___________ Vertices:___________ Foci:__________ Foci:__________ Asymptotes:________ Asymptotes:________ 23. (𝑦+2)2 25 − (𝑥−3)2 16 =1 24. (𝑦+1)2 25 (𝑥−2)2 25 − – (𝑥−2)2 9 (𝑦+3)2 16 =1 =1 Center: ___________ Center: ___________ Vertices:___________ Vertices:___________ Foci:__________ Foci:__________ Asymptotes:________ Asymptotes:________ Find the standard form of the equation of each hyperbola. 25. Foci (0, ±4), vertices (0, ±2) 26. Vertices (±4, 0), Asymptotes: 𝑦 = ±3𝑥 27. Endpoints of transverse axis: (±6, 0) Asymptotes: 𝑦 = ±2𝑥 28. Foci (0, ±3), length of transverse axis 2 29. 30. Graph the parabola and identify the vertex, directrix, and focus. 31. 𝑦 2 = −12𝑥 32. 𝑥 2 = 8𝑦 Vertex: _______ Vertex: _______ Dir: _______ Dir: ____________ Focus: ___________ Focus:_________ 33. 6(𝑥 + 1)2 + 12(𝑦 − 3) = 0 34. 𝑦 2 − 12(𝑥 + 2) = 0 Vertex: _______ Vertex: _______ Dir: ___________ Dir: __________ Focus: ________ Focus:________ 35. (𝑥 + 2)2 = −8(𝑦 + 2) 36. (𝑦 − 1)2 = −8𝑥 Vertex: _______ Vertex: _______ Dir: _______ Dir: ____________ Focus: ___________ Focus:_________ Write an equation in standard form for the parabola satisfying the given conditions. 37. Focus: (9, 0); Directrix: 𝑥 = −9 38. Focus: (−10, 0); Directrix: 𝑥 = 10 39. Vertex: (5, −2); Focus (7, −2) 40. Focus: (2, 4); Directrix: 𝑥 = −4 41. 42. Convert the equation to standard form by completing the square. Then identify what type of conic section the equation represents. If it is a circle, ellipse, or hyperbola, then name its center. If it is a parabola, then name its vertex. 43. 𝑥 2 + 6𝑥 + 8𝑦 + 1 = 0 44. 9𝑦 2 − 4𝑥 2 − 18𝑦 + 24𝑥 − 63 = 0 45. 4𝑥 2 + 36𝑦 − 32𝑥 + 9𝑦 2 + 64 = 0 46. 9𝑥 2 + 16𝑦 2 − 18𝑥 + 64𝑦 − 71 = 0 47. 4𝑥 2 − 𝑦 2 + 32𝑥 + 6𝑦 + 39 = 0 48. 𝑥 2 + 𝑦 2 − 6𝑥 + 8𝑦 = 1 49. 4𝑥 2 + 4𝑦 2 − 24𝑥 + 32𝑦 − 4 = 0 50. 𝑦 2 + 8𝑦 − 4𝑥 + 8 = 0 Answers to Conics Test Review Pre-Calculus Name:________________________________ Hour:_____ 5 1. Center: (5, −3); Radius: 3 2. Center: (−3, 0); Radius: 3 3. 𝑥 2 + 𝑦 2 = 25 4. (𝑥 + 4)2 + (𝑦 + 3)2 = 13 5. (𝑥 + 3)2 + (𝑦 − 5)2 = 26 6. (𝑥 − 4)2 + (𝑦 + 3)2 = 9 7. C: (0, 0); V: (0, ±5); F: (0, ±4) 8. C: (0, 0); V: (±3, 0); F: (±√5, 0) 9. C: (0, 0); V: (±6, 0); F: (±3√3, 0) 10. C: (−1, 3); V: (−1, 5), (−1, 1); F: (−1, 3 ± √3) 11. C: (1, −2); V: (1, 1), (1, −5); F: (1, −2 ± √5) 12. C: (−4, −3); V: (−4, 3), (−4, −9); F: (−4, −3 ± √35) 15. 𝑥2 36 𝑦2 + 11 = 1 16. (𝑥−5)2 9 + (𝑦−2)2 1 𝑥2 16 13. (𝑥+1)2 9 + 𝑦2 25 14. =1 𝑦2 (𝑥−2)2 9 + (𝑦+1)2 4 (𝑦−3)2 4 =1 17. 3 20. C: (0, 0); V: (±2, 0); F: (±2√5, 0); A: ± 2 + 25 = 1 18. (𝑥+1)2 36 + =1 =1 4 19. C: (0, 0); V: (0, ±3); F: (0, ±5); A: ± 4 2 21. C: (1, −2); V: (−2, −2), (4, −2); F: (1 ± √13, −2); A: ± 3 5 22. C: (2, −1); V: (2, 4), (2, −6); F: (2, −1 ± √34); A: ± 3 5 23. C: (3, −2); V: (3, 3), (3, −7); F: (3, −2 ± √41); A: ± 4 4 24. C: (2, −3); V: (−3, −3), (7, −3); F: (2 ± √41, −3); A: ± 5 27. 𝑥2 36 𝑦2 − 144 = 1 28. 𝑦2 1 − 𝑥2 8 =1 29. (𝑥−2)2 4 − 25. (𝑦−1)2 9 =1 𝑦2 4 𝑥2 30. 31. V: (0, 0); D: 𝑥 = 3; F: (−3, 0) 32. V: (0, 0); D: 𝑦 = −2; F: (0, 2) 34. V: (−2, 0); D: 𝑥 = −5; F: (1, 0) 35. V: (−2, −2); D: 𝑦 = 0; F: (−2, −4) 𝑦2 9 − 𝑥2 1 36. V: (0, 1); D: 𝑥 = 2; F: (−2, 1) 41. (𝑦 − 1)2 = 8𝑥 42. 𝑥 2 = −12(𝑦 − 1) 43. (𝑥 + 3)2 = −8(𝑦 − 1); Parabola; V: (−3, 1) 46. 4 (𝑥−1)2 16 − + (𝑥−3)2 9 (𝑦+2)2 9 = 1; Hyperbola; C:(3, 1) 45. = 1; Ellipse; C:(1, −2) 47. 48. (𝑥 − 3)2 + (𝑦 + 4)2 = 26; Circle; C: (3, −4) 50. (𝑦 + 4)2 = 4(𝑥 + 2); Parabola; V: (−2, −4) (𝑥−4)2 9 + (𝑥+4)2 4 (𝑦+2)2 − 4 (𝑦−3)2 16 𝑦2 − 144 = 1 33.V: (−1, 3); D: 𝑦 = 3.5; F: (−1, 2.5) 38. 𝑦 2 = −40𝑥 (𝑦−1)2 𝑥2 16 =1 37. 𝑦 2 = 36𝑥 44. 39. (𝑦 + 2)2 = 8(𝑥 − 5) 26. − 12 = 1 40. (𝑦 − 4)2 = 12(𝑥 − 2) = 1; Ellipse; C: (4, −2) = 1; Hyperbola; C:(−4, 3) 49. (𝑥 − 3)2 + (𝑦 + 4)2 = 26; Circle; C: (3, −4)