1. Explain how eccentricity determines which conic section is given.\n\n2.\u00a0If a conic section is written as a polar equation, what must be true of the denominator?\n\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?\n\n4.\u00a0If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?\n\n5. What do we know about the focus\/foci of a conic section if it is written as a polar equation?\n\nFor the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.\n\n6. [latex]r=\\frac{6}{1 - 2\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n7. [latex]r=\\frac{3}{4 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n8.\u00a0[latex]r=\\frac{8}{4 - 3\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n9. [latex]r=\\frac{5}{1+2\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n10.\u00a0[latex]r=\\frac{16}{4+3\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n11. [latex]r=\\frac{3}{10+10\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n12.\u00a0[latex]r=\\frac{2}{1-\\cos \\text{ }\\theta }[\/latex]\n\n13. [latex]r=\\frac{4}{7+2\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n14.\u00a0[latex]r\\left(1-\\cos \\text{ }\\theta \\right)=3[\/latex]\n\n15. [latex]r\\left(3+5\\sin \\text{ }\\theta \\right)=11[\/latex]\n\n16.\u00a0[latex]r\\left(4 - 5\\sin \\text{ }\\theta \\right)=1[\/latex]\n\n17. [latex]r\\left(7+8\\cos \\text{ }\\theta \\right)=7[\/latex]\n\nFor the following exercises, convert the polar equation of a conic section to a rectangular equation.\n\n18. [latex]r=\\frac{4}{1+3\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n19. [latex]r=\\frac{2}{5 - 3\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n20.\u00a0[latex]r=\\frac{8}{3 - 2\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n21. [latex]r=\\frac{3}{2+5\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n22.\u00a0[latex]r=\\frac{4}{2+2\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n23. [latex]r=\\frac{3}{8 - 8\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n24.\u00a0[latex]r=\\frac{2}{6+7\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n25. [latex]r=\\frac{5}{5 - 11\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n26.\u00a0[latex]r\\left(5+2\\text{ }\\cos \\text{ }\\theta \\right)=6[\/latex]\n\n27. [latex]r\\left(2-\\cos \\text{ }\\theta \\right)=1[\/latex]\n\n28.\u00a0[latex]r\\left(2.5 - 2.5\\text{ }\\sin \\text{ }\\theta \\right)=5[\/latex]\n\n29. [latex]r=\\frac{6\\sec \\text{ }\\theta }{-2+3\\text{ }\\sec \\text{ }\\theta }[\/latex]\n\n30.\u00a0[latex]r=\\frac{6\\csc \\text{ }\\theta }{3+2\\text{ }\\csc \\text{ }\\theta }[\/latex]\n\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.\n\n31. [latex]r=\\frac{5}{2+\\cos \\text{ }\\theta }[\/latex]\n\n32. [latex]r=\\frac{2}{3+3\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n33. [latex]r=\\frac{10}{5 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n34. [latex]r=\\frac{3}{1+2\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n35. [latex]r=\\frac{8}{4 - 5\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n36. [latex]r=\\frac{3}{4 - 4\\text{ }\\cos \\text{ }\\theta }[\/latex]\n\n37. [latex]r=\\frac{2}{1-\\sin \\text{ }\\theta }[\/latex]\n\n38. [latex]r=\\frac{6}{3+2\\text{ }\\sin \\text{ }\\theta }[\/latex]\n\n39. [latex]r\\left(1+\\cos \\text{ }\\theta \\right)=5[\/latex]\n\n40. [latex]r\\left(3 - 4\\sin \\text{ }\\theta \\right)=9[\/latex]\n\n41. [latex]r\\left(3 - 2\\sin \\text{ }\\theta \\right)=6[\/latex]\n\n42. [latex]r\\left(6 - 4\\cos \\text{ }\\theta \\right)=5[\/latex]\n\nFor the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.\n\n43. Directrix: [latex]x=4;e=\\frac{1}{5}[\/latex]\n\n44.\u00a0Directrix: [latex]x=-4;e=5[\/latex]\n\n45. Directrix: [latex]y=2;e=2[\/latex]\n\n46.\u00a0Directrix: [latex]y=-2;e=\\frac{1}{2}[\/latex]\n\n47. Directrix: [latex]x=1;e=1[\/latex]\n\n48.\u00a0Directrix: [latex]x=-1;e=1[\/latex]\n\n49. Directrix: [latex]x=-\\frac{1}{4};e=\\frac{7}{2}[\/latex]\n\n50.\u00a0Directrix: [latex]y=\\frac{2}{5};e=\\frac{7}{2}[\/latex]\n\n51. Directrix: [latex]y=4;e=\\frac{3}{2}[\/latex]\n\n52.\u00a0Directrix: [latex]x=-2;e=\\frac{8}{3}[\/latex]\n\n53. Directrix: [latex]x=-5;e=\\frac{3}{4}[\/latex]\n\n54.\u00a0Directrix: [latex]y=2;e=2.5[\/latex]\n\n55. Directrix: [latex]x=-3;e=\\frac{1}{3}[\/latex]\n\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].\n\n56. [latex]xy=2[\/latex]\n\n57. [latex]{x}^{2}+xy+{y}^{2}=4[\/latex]\n\n58.\u00a0[latex]2{x}^{2}+4xy+2{y}^{2}=9[\/latex]\n\n59. [latex]16{x}^{2}+24xy+9{y}^{2}=4[\/latex]\n\n60.\u00a0[latex]2xy+y=1[\/latex]<\/p>","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 Candela Citations<\/h3>\n\t\t\t\t\t