{"id":1444,"date":"2023-06-05T14:51:48","date_gmt":"2023-06-05T14:51:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-conic-sections-in-polar-coordinates\/"},"modified":"2023-06-05T14:51:48","modified_gmt":"2023-06-05T14:51:48","slug":"solutions-for-conic-sections-in-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-conic-sections-in-polar-coordinates\/","title":{"raw":"Solutions 69: Conic Sections in Polar Coordinates","rendered":"Solutions 69: Conic Sections in Polar Coordinates"},"content":{"raw":"\nSolutions to Odd-Numbered Exercises<\/h2>\n1. If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.\n\n3. The directrix will be parallel to the polar axis.\n\n5. One of the foci will be located at the origin.\n\n7. Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{4}[\/latex] units below the pole.\n\n9. Hyperbola with [latex]e=2[\/latex] and directrix [latex]\\frac{5}{2}[\/latex] units above the pole.\n\n11. Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{10}[\/latex] units to the right of the pole.\n\n13. Ellipse with [latex]e=\\frac{2}{7}[\/latex] and directrix [latex]2[\/latex] units to the right of the pole.\n\n15. Hyperbola with [latex]e=\\frac{5}{3}[\/latex] and directrix [latex]\\frac{11}{5}[\/latex] units above the pole.\n\n17. Hyperbola with [latex]e=\\frac{8}{7}[\/latex] and directrix [latex]\\frac{7}{8}[\/latex] units to the right of the pole.\n\n19. [latex]25{x}^{2}+16{y}^{2}-12y - 4=0[\/latex]\n\n21. [latex]21{x}^{2}-4{y}^{2}-30x+9=0[\/latex]\n\n23. [latex]64{y}^{2}=48x+9[\/latex]\n\n25. [latex]96{y}^{2}-25{x}^{2}+110y+25=0[\/latex]\n\n27. [latex]3{x}^{2}+4{y}^{2}-2x - 1=0[\/latex]\n\n29. [latex]5{x}^{2}+9{y}^{2}-24x - 36=0[\/latex]\n\n31.\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183139\/CNX_Precalc_Figure_10_05_2012.jpg\")
\n\n33.\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183141\/CNX_Precalc_Figure_10_05_2032.jpg\")
\n\n35.\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183142\/CNX_Precalc_Figure_10_05_2052.jpg\")
\n\n37.\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183145\/CNX_Precalc_Figure_10_05_2072.jpg\")
\n\n39.\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183147\/CNX_Precalc_Figure_10_05_2092.jpg\")
\n\n41.\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183150\/CNX_Precalc_Figure_10_05_2112.jpg\")
\n\n43. [latex]r=\\frac{4}{5+\\cos \\theta }[\/latex]\n\n45. [latex]r=\\frac{4}{1+2\\sin \\theta }[\/latex]\n\n47. [latex]r=\\frac{1}{1+\\cos \\theta }[\/latex]\n\n49. [latex]r=\\frac{7}{8 - 28\\cos \\theta }[\/latex]\n\n51. [latex]r=\\frac{12}{2+3\\sin \\theta }[\/latex]\n\n53. [latex]r=\\frac{15}{4 - 3\\cos \\theta }[\/latex]\n\n55. [latex]r=\\frac{3}{3 - 3\\cos \\theta }[\/latex]\n\n57. [latex]r=\\pm \\frac{2}{\\sqrt{1+\\sin \\theta \\cos \\theta }}[\/latex]\n\n59. [latex]r=\\pm \\frac{2}{4\\cos \\theta +3\\sin \\theta }[\/latex]\n","rendered":"Solutions to Odd-Numbered Exercises<\/h2>\n
1. If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.<\/p>\n
3. The directrix will be parallel to the polar axis.<\/p>\n
5. One of the foci will be located at the origin.<\/p>\n
7. Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{4}[\/latex] units below the pole.<\/p>\n
9. Hyperbola with [latex]e=2[\/latex] and directrix [latex]\\frac{5}{2}[\/latex] units above the pole.<\/p>\n
11. Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{10}[\/latex] units to the right of the pole.<\/p>\n
13. Ellipse with [latex]e=\\frac{2}{7}[\/latex] and directrix [latex]2[\/latex] units to the right of the pole.<\/p>\n
15. Hyperbola with [latex]e=\\frac{5}{3}[\/latex] and directrix [latex]\\frac{11}{5}[\/latex] units above the pole.<\/p>\n
17. Hyperbola with [latex]e=\\frac{8}{7}[\/latex] and directrix [latex]\\frac{7}{8}[\/latex] units to the right of the pole.<\/p>\n
19. [latex]25{x}^{2}+16{y}^{2}-12y - 4=0[\/latex]<\/p>\n
21. [latex]21{x}^{2}-4{y}^{2}-30x+9=0[\/latex]<\/p>\n
23. [latex]64{y}^{2}=48x+9[\/latex]<\/p>\n
25. [latex]96{y}^{2}-25{x}^{2}+110y+25=0[\/latex]<\/p>\n
27. [latex]3{x}^{2}+4{y}^{2}-2x - 1=0[\/latex]<\/p>\n
29. [latex]5{x}^{2}+9{y}^{2}-24x - 36=0[\/latex]<\/p>\n
31.
\n<\/p>\n
33.
\n<\/p>\n
35.
\n<\/p>\n
37.
\n<\/p>\n
39.
\n<\/p>\n
41.
\n<\/p>\n
43. [latex]r=\\frac{4}{5+\\cos \\theta }[\/latex]<\/p>\n
45. [latex]r=\\frac{4}{1+2\\sin \\theta }[\/latex]<\/p>\n
47. [latex]r=\\frac{1}{1+\\cos \\theta }[\/latex]<\/p>\n
49. [latex]r=\\frac{7}{8 - 28\\cos \\theta }[\/latex]<\/p>\n
51. [latex]r=\\frac{12}{2+3\\sin \\theta }[\/latex]<\/p>\n
53. [latex]r=\\frac{15}{4 - 3\\cos \\theta }[\/latex]<\/p>\n
55. [latex]r=\\frac{3}{3 - 3\\cos \\theta }[\/latex]<\/p>\n
57. [latex]r=\\pm \\frac{2}{\\sqrt{1+\\sin \\theta \\cos \\theta }}[\/latex]<\/p>\n
59. [latex]r=\\pm \\frac{2}{4\\cos \\theta +3\\sin \\theta }[\/latex]<\/p>\n\n\t\t\t