{"id":15776,"date":"2019-09-05T19:46:01","date_gmt":"2019-09-05T19:46:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15776"},"modified":"2025-02-05T05:24:53","modified_gmt":"2025-02-05T05:24:53","slug":"problem-set-69-conic-sections-in-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-69-conic-sections-in-polar-coordinates\/","title":{"raw":"Problem Set 69: Conic Sections in Polar Coordinates","rendered":"Problem Set 69: Conic Sections in Polar Coordinates"},"content":{"raw":"1. Explain how eccentricity determines which conic section is given.\r\n\r\n2.\u00a0If a conic section is written as a polar equation, what must be true of the denominator?\r\n\r\n3. If a conic section is written as a polar equation, and the denominator involves [latex]\\sin \\text{ }\\theta [\/latex], what conclusion can be drawn about the directrix?\r\n\r\n4.\u00a0If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?\r\n\r\n5. What do we know about the focus\/foci of a conic section if it is written as a polar equation?\r\n\r\nFor the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.\r\n\r\n6. [latex]r=\\frac{6}{1 - 2\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n7. [latex]r=\\frac{3}{4 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n8.\u00a0[latex]r=\\frac{8}{4 - 3\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n9. [latex]r=\\frac{5}{1+2\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n10.\u00a0[latex]r=\\frac{16}{4+3\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n11. [latex]r=\\frac{3}{10+10\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n12.\u00a0[latex]r=\\frac{2}{1-\\cos \\text{ }\\theta }[\/latex]\r\n\r\n13. [latex]r=\\frac{4}{7+2\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n14.\u00a0[latex]r\\left(1-\\cos \\text{ }\\theta \\right)=3[\/latex]\r\n\r\n15. [latex]r\\left(3+5\\sin \\text{ }\\theta \\right)=11[\/latex]\r\n\r\n16.\u00a0[latex]r\\left(4 - 5\\sin \\text{ }\\theta \\right)=1[\/latex]\r\n\r\n17. [latex]r\\left(7+8\\cos \\text{ }\\theta \\right)=7[\/latex]\r\n\r\nFor the following exercises, convert the polar equation of a conic section to a rectangular equation.\r\n\r\n18. [latex]r=\\frac{4}{1+3\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n19. [latex]r=\\frac{2}{5 - 3\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n20.\u00a0[latex]r=\\frac{8}{3 - 2\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n21. [latex]r=\\frac{3}{2+5\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n22.\u00a0[latex]r=\\frac{4}{2+2\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n23. [latex]r=\\frac{3}{8 - 8\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n24.\u00a0[latex]r=\\frac{2}{6+7\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n25. [latex]r=\\frac{5}{5 - 11\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n26.\u00a0[latex]r\\left(5+2\\text{ }\\cos \\text{ }\\theta \\right)=6[\/latex]\r\n\r\n27. [latex]r\\left(2-\\cos \\text{ }\\theta \\right)=1[\/latex]\r\n\r\n28.\u00a0[latex]r\\left(2.5 - 2.5\\text{ }\\sin \\text{ }\\theta \\right)=5[\/latex]\r\n\r\n29. [latex]r=\\frac{6\\sec \\text{ }\\theta }{-2+3\\text{ }\\sec \\text{ }\\theta }[\/latex]\r\n\r\n30.\u00a0[latex]r=\\frac{6\\csc \\text{ }\\theta }{3+2\\text{ }\\csc \\text{ }\\theta }[\/latex]\r\n\r\nFor the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.\r\n\r\n31. [latex]r=\\frac{5}{2+\\cos \\text{ }\\theta }[\/latex]\r\n\r\n32. [latex]r=\\frac{2}{3+3\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n33. [latex]r=\\frac{10}{5 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n34. [latex]r=\\frac{3}{1+2\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n35. [latex]r=\\frac{8}{4 - 5\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n36. [latex]r=\\frac{3}{4 - 4\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n37. [latex]r=\\frac{2}{1-\\sin \\text{ }\\theta }[\/latex]\r\n\r\n38. [latex]r=\\frac{6}{3+2\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n39. [latex]r\\left(1+\\cos \\text{ }\\theta \\right)=5[\/latex]\r\n\r\n40. [latex]r\\left(3 - 4\\sin \\text{ }\\theta \\right)=9[\/latex]\r\n\r\n41. [latex]r\\left(3 - 2\\sin \\text{ }\\theta \\right)=6[\/latex]\r\n\r\n42. [latex]r\\left(6 - 4\\cos \\text{ }\\theta \\right)=5[\/latex]\r\n\r\nFor the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.\r\n\r\n43. Directrix: [latex]x=4;e=\\frac{1}{5}[\/latex]\r\n\r\n44.\u00a0Directrix: [latex]x=-4;e=5[\/latex]\r\n\r\n45. Directrix: [latex]y=2;e=2[\/latex]\r\n\r\n46.\u00a0Directrix: [latex]y=-2;e=\\frac{1}{2}[\/latex]\r\n\r\n47. Directrix: [latex]x=1;e=1[\/latex]\r\n\r\n48.\u00a0Directrix: [latex]x=-1;e=1[\/latex]\r\n\r\n49. Directrix: [latex]x=-\\frac{1}{4};e=\\frac{7}{2}[\/latex]\r\n\r\n50.\u00a0Directrix: [latex]y=\\frac{2}{5};e=\\frac{7}{2}[\/latex]\r\n\r\n51. Directrix: [latex]y=4;e=\\frac{3}{2}[\/latex]\r\n\r\n52.\u00a0Directrix: [latex]x=-2;e=\\frac{8}{3}[\/latex]\r\n\r\n53. Directrix: [latex]x=-5;e=\\frac{3}{4}[\/latex]\r\n\r\n54.\u00a0Directrix: [latex]y=2;e=2.5[\/latex]\r\n\r\n55. Directrix: [latex]x=-3;e=\\frac{1}{3}[\/latex]\r\n\r\nEquations of conics with an [latex]xy[\/latex] term have rotated graphs. For the following exercises, express each equation in polar form with [latex]r[\/latex] as a function of [latex]\\theta [\/latex].\r\n\r\n56. [latex]xy=2[\/latex]\r\n\r\n57. [latex]{x}^{2}+xy+{y}^{2}=4[\/latex]\r\n\r\n58.\u00a0[latex]2{x}^{2}+4xy+2{y}^{2}=9[\/latex]\r\n\r\n59. [latex]16{x}^{2}+24xy+9{y}^{2}=4[\/latex]\r\n\r\n60.\u00a0[latex]2xy+y=1[\/latex]","rendered":"

1. Explain how eccentricity determines which conic section is given.<\/p>\n

2.\u00a0If a conic section is written as a polar equation, what must be true of the denominator?<\/p>\n

3. If a conic section is written as a polar equation, and the denominator involves [latex]\\sin \\text{ }\\theta[\/latex], what conclusion can be drawn about the directrix?<\/p>\n

4.\u00a0If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?<\/p>\n

5. What do we know about the focus\/foci of a conic section if it is written as a polar equation?<\/p>\n

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.<\/p>\n

6. [latex]r=\\frac{6}{1 - 2\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

7. [latex]r=\\frac{3}{4 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

8.\u00a0[latex]r=\\frac{8}{4 - 3\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

9. [latex]r=\\frac{5}{1+2\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

10.\u00a0[latex]r=\\frac{16}{4+3\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

11. [latex]r=\\frac{3}{10+10\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

12.\u00a0[latex]r=\\frac{2}{1-\\cos \\text{ }\\theta }[\/latex]<\/p>\n

13. [latex]r=\\frac{4}{7+2\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

14.\u00a0[latex]r\\left(1-\\cos \\text{ }\\theta \\right)=3[\/latex]<\/p>\n

15. [latex]r\\left(3+5\\sin \\text{ }\\theta \\right)=11[\/latex]<\/p>\n

16.\u00a0[latex]r\\left(4 - 5\\sin \\text{ }\\theta \\right)=1[\/latex]<\/p>\n

17. [latex]r\\left(7+8\\cos \\text{ }\\theta \\right)=7[\/latex]<\/p>\n

For the following exercises, convert the polar equation of a conic section to a rectangular equation.<\/p>\n

18. [latex]r=\\frac{4}{1+3\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

19. [latex]r=\\frac{2}{5 - 3\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

20.\u00a0[latex]r=\\frac{8}{3 - 2\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

21. [latex]r=\\frac{3}{2+5\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

22.\u00a0[latex]r=\\frac{4}{2+2\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

23. [latex]r=\\frac{3}{8 - 8\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

24.\u00a0[latex]r=\\frac{2}{6+7\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

25. [latex]r=\\frac{5}{5 - 11\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

26.\u00a0[latex]r\\left(5+2\\text{ }\\cos \\text{ }\\theta \\right)=6[\/latex]<\/p>\n

27. [latex]r\\left(2-\\cos \\text{ }\\theta \\right)=1[\/latex]<\/p>\n

28.\u00a0[latex]r\\left(2.5 - 2.5\\text{ }\\sin \\text{ }\\theta \\right)=5[\/latex]<\/p>\n

29. [latex]r=\\frac{6\\sec \\text{ }\\theta }{-2+3\\text{ }\\sec \\text{ }\\theta }[\/latex]<\/p>\n

30.\u00a0[latex]r=\\frac{6\\csc \\text{ }\\theta }{3+2\\text{ }\\csc \\text{ }\\theta }[\/latex]<\/p>\n

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.<\/p>\n

31. [latex]r=\\frac{5}{2+\\cos \\text{ }\\theta }[\/latex]<\/p>\n

32. [latex]r=\\frac{2}{3+3\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

33. [latex]r=\\frac{10}{5 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

34. [latex]r=\\frac{3}{1+2\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

35. [latex]r=\\frac{8}{4 - 5\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

36. [latex]r=\\frac{3}{4 - 4\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

37. [latex]r=\\frac{2}{1-\\sin \\text{ }\\theta }[\/latex]<\/p>\n

38. [latex]r=\\frac{6}{3+2\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

39. [latex]r\\left(1+\\cos \\text{ }\\theta \\right)=5[\/latex]<\/p>\n

40. [latex]r\\left(3 - 4\\sin \\text{ }\\theta \\right)=9[\/latex]<\/p>\n

41. [latex]r\\left(3 - 2\\sin \\text{ }\\theta \\right)=6[\/latex]<\/p>\n

42. [latex]r\\left(6 - 4\\cos \\text{ }\\theta \\right)=5[\/latex]<\/p>\n

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.<\/p>\n

43. Directrix: [latex]x=4;e=\\frac{1}{5}[\/latex]<\/p>\n

44.\u00a0Directrix: [latex]x=-4;e=5[\/latex]<\/p>\n

45. Directrix: [latex]y=2;e=2[\/latex]<\/p>\n

46.\u00a0Directrix: [latex]y=-2;e=\\frac{1}{2}[\/latex]<\/p>\n

47. Directrix: [latex]x=1;e=1[\/latex]<\/p>\n

48.\u00a0Directrix: [latex]x=-1;e=1[\/latex]<\/p>\n

49. Directrix: [latex]x=-\\frac{1}{4};e=\\frac{7}{2}[\/latex]<\/p>\n

50.\u00a0Directrix: [latex]y=\\frac{2}{5};e=\\frac{7}{2}[\/latex]<\/p>\n

51. Directrix: [latex]y=4;e=\\frac{3}{2}[\/latex]<\/p>\n

52.\u00a0Directrix: [latex]x=-2;e=\\frac{8}{3}[\/latex]<\/p>\n

53. Directrix: [latex]x=-5;e=\\frac{3}{4}[\/latex]<\/p>\n

54.\u00a0Directrix: [latex]y=2;e=2.5[\/latex]<\/p>\n

55. Directrix: [latex]x=-3;e=\\frac{1}{3}[\/latex]<\/p>\n

Equations of conics with an [latex]xy[\/latex] term have rotated graphs. For the following exercises, express each equation in polar form with [latex]r[\/latex] as a function of [latex]\\theta[\/latex].<\/p>\n

56. [latex]xy=2[\/latex]<\/p>\n

57. [latex]{x}^{2}+xy+{y}^{2}=4[\/latex]<\/p>\n

58.\u00a0[latex]2{x}^{2}+4xy+2{y}^{2}=9[\/latex]<\/p>\n

59. [latex]16{x}^{2}+24xy+9{y}^{2}=4[\/latex]<\/p>\n

60.\u00a0[latex]2xy+y=1[\/latex]<\/p>\n\n\t\t\t

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