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Study Guides > Precalculus II

Solutions for The Ellipse

Solutions to Try Its

1. x2+y216=1{x}^{2}+\frac{{y}^{2}}{16}=1 2. (x−1)216+(y−3)24=1\frac{{\left(x - 1\right)}^{2}}{16}+\frac{{\left(y - 3\right)}^{2}}{4}=1 3. center: (0,0)\left(0,0\right); vertices: (±6,0)\left(\pm 6,0\right); co-vertices: (0,±2)\left(0,\pm 2\right); foci: (±42,0)\left(\pm 4\sqrt{2},0\right) 4. Standard form: x216+y249=1\frac{{x}^{2}}{16}+\frac{{y}^{2}}{49}=1; center: (0,0)\left(0,0\right); vertices: (0,±7)\left(0,\pm 7\right); co-vertices: (±4,0)\left(\pm 4,0\right); foci: (0,±33)\left(0,\pm \sqrt{33}\right) 5. Center: (4,2)\left(4,2\right); vertices: (−2,2)\left(-2,2\right) and (10,2)\left(10,2\right); co-vertices: (4,2−25)\left(4,2 - 2\sqrt{5}\right) and (4,2+25)\left(4,2+2\sqrt{5}\right); foci: (0,2)\left(0,2\right) and (8,2)\left(8,2\right) 6. (x−3)24+(y+1)216=1\frac{{\left(x - 3\right)}^{2}}{4}+\frac{{\left(y+1\right)}^{2}}{16}=1; center: (3,−1)\left(3,-1\right); vertices: (3,−5)\left(3,-\text{5}\right) and (3,3)\left(3,\text{3}\right); co-vertices: (1,−1)\left(1,-1\right) and (5,−1)\left(5,-1\right); foci: (3,−1−23)\left(3,-\text{1}-2\sqrt{3}\right) and (3,−1+23)\left(3,-\text{1+}2\sqrt{3}\right) 7. a. x257,600+y225,600=1\frac{{x}^{2}}{57,600}+\frac{{y}^{2}}{25,600}=1 b. The people are standing 358 feet apart.

Solutions to Odd-Numbered Exercises

1. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant. 3. This special case would be a circle. 5. It is symmetric about the x-axis, y-axis, and the origin. 7. yes; x232+y222=1\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1 9. yes; x2(12)2+y2(13)2=1\frac{{x}^{2}}{{\left(\frac{1}{2}\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1 11. x222+y272=1\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{7}^{2}}=1; Endpoints of major axis (0,7)\left(0,7\right) and (0,−7)\left(0,-7\right). Endpoints of minor axis (2,0)\left(2,0\right) and (−2,0)\left(-2,0\right). Foci at (0,35),(0,−35)\left(0,3\sqrt{5}\right),\left(0,-3\sqrt{5}\right). 13. x2(1)2+y2(13)2=1\frac{{x}^{2}}{{\left(1\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1; Endpoints of major axis (1,0)\left(1,0\right) and (−1,0)\left(-1,0\right). Endpoints of minor axis (0,13),(0,−13)\left(0,\frac{1}{3}\right),\left(0,-\frac{1}{3}\right). Foci at (223,0),(−223,0)\left(\frac{2\sqrt{2}}{3},0\right),\left(-\frac{2\sqrt{2}}{3},0\right). 15. (x−2)272+(y−4)252=1\frac{{\left(x - 2\right)}^{2}}{{7}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{5}^{2}}=1; Endpoints of major axis (9,4),(−5,4)\left(9,4\right),\left(-5,4\right). Endpoints of minor axis (2,9),(2,−1)\left(2,9\right),\left(2,-1\right). Foci at (2+26,4),(2−26,4)\left(2+2\sqrt{6},4\right),\left(2 - 2\sqrt{6},4\right). 17. (x+5)222+(y−7)232=1\frac{{\left(x+5\right)}^{2}}{{2}^{2}}+\frac{{\left(y - 7\right)}^{2}}{{3}^{2}}=1; Endpoints of major axis (−5,10),(−5,4)\left(-5,10\right),\left(-5,4\right). Endpoints of minor axis (−3,7),(−7,7)\left(-3,7\right),\left(-7,7\right). Foci at (−5,7+5),(−5,7−5)\left(-5,7+\sqrt{5}\right),\left(-5,7-\sqrt{5}\right). 19. (x−1)232+(y−4)222=1\frac{{\left(x - 1\right)}^{2}}{{3}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{2}^{2}}=1; Endpoints of major axis (4,4),(−2,4)\left(4,4\right),\left(-2,4\right). Endpoints of minor axis (1,6),(1,2)\left(1,6\right),\left(1,2\right). Foci at (1+5,4),(1−5,4)\left(1+\sqrt{5},4\right),\left(1-\sqrt{5},4\right). 21. (x−3)2(32)2+(y−5)2(2)2=1\frac{{\left(x - 3\right)}^{2}}{{\left(3\sqrt{2}\right)}^{2}}+\frac{{\left(y - 5\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}=1; Endpoints of major axis (3+32,5),(3−32,5)\left(3+3\sqrt{2},5\right),\left(3 - 3\sqrt{2},5\right). Endpoints of minor axis (3,5+2),(3,5−2)\left(3,5+\sqrt{2}\right),\left(3,5-\sqrt{2}\right). Foci at (7,5),(−1,5)\left(7,5\right),\left(-1,5\right). 23. (x+5)2(5)2+(y−2)2(2)2=1\frac{{\left(x+5\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y - 2\right)}^{2}}{{\left(2\right)}^{2}}=1; Endpoints of major axis (0,2),(−10,2)\left(0,2\right),\left(-10,2\right). Endpoints of minor axis (−5,4),(−5,0)\left(-5,4\right),\left(-5,0\right). Foci at (−5+21,2),(−5−21,2)\left(-5+\sqrt{21},2\right),\left(-5-\sqrt{21},2\right). 25. (x+3)2(5)2+(y+4)2(2)2=1\frac{{\left(x+3\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y+4\right)}^{2}}{{\left(2\right)}^{2}}=1; Endpoints of major axis (2,−4),(−8,−4)\left(2,-4\right),\left(-8,-4\right). Endpoints of minor axis (−3,−2),(−3,−6)\left(-3,-2\right),\left(-3,-6\right). Foci at (−3+21,−4),(−3−21,−4)\left(-3+\sqrt{21},-4\right),\left(-3-\sqrt{21},-4\right). 27. Foci (−3,−1+11),(−3,−1−11)\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right) 29. Focus (0,0)\left(0,0\right) 31. Foci (−10,30),(−10,−30)\left(-10,30\right),\left(-10,-30\right) 33. Center (0,0)\left(0,0\right), Vertices (4,0),(−4,0),(0,3),(0,−3)\left(4,0\right),\left(-4,0\right),\left(0,3\right),\left(0,-3\right), Foci (7,0),(−7,0)\left(\sqrt{7},0\right),\left(-\sqrt{7},0\right) 35. Center (0,0)\left(0,0\right), Vertices (19,0),(−19,0),(0,17),(0,−17)\left(\frac{1}{9},0\right),\left(-\frac{1}{9},0\right),\left(0,\frac{1}{7}\right),\left(0,-\frac{1}{7}\right), Foci (0,4263),(0,−4263)\left(0,\frac{4\sqrt{2}}{63}\right),\left(0,-\frac{4\sqrt{2}}{63}\right) 37. Center (−3,3)\left(-3,3\right), Vertices (0,3),(−6,3),(−3,0),(−3,6)\left(0,3\right),\left(-6,3\right),\left(-3,0\right),\left(-3,6\right), Focus (−3,3)\left(-3,3\right) Note that this ellipse is a circle. The circle has only one focus, which coincides with the center. 39. Center (1,1)\left(1,1\right), Vertices (5,1),(−3,1),(1,3),(1,−1)\left(5,1\right),\left(-3,1\right),\left(1,3\right),\left(1,-1\right), Foci (1,1+43),(1,1−43)\left(1,1+4\sqrt{3}\right),\left(1,1 - 4\sqrt{3}\right) 41. Center (−4,5)\left(-4,5\right), Vertices (−2,5),(−6,4),(−4,6),(−4,4)\left(-2,5\right),\left(-6,4\right),\left(-4,6\right),\left(-4,4\right), Foci (−4+3,5),(−4−3,5)\left(-4+\sqrt{3},5\right),\left(-4-\sqrt{3},5\right) 43. Center (−2,1)\left(-2,1\right), Vertices (0,1),(−4,1),(−2,5),(−2,−3)\left(0,1\right),\left(-4,1\right),\left(-2,5\right),\left(-2,-3\right), Foci (−2,1+23),(−2,1−23)\left(-2,1+2\sqrt{3}\right),\left(-2,1 - 2\sqrt{3}\right) 45. Center (−2,−2)\left(-2,-2\right), Vertices (0,−2),(−4,−2),(−2,0),(−2,−4)\left(0,-2\right),\left(-4,-2\right),\left(-2,0\right),\left(-2,-4\right), Focus (−2,−2)\left(-2,-2\right) 47. x225+y229=1\frac{{x}^{2}}{25}+\frac{{y}^{2}}{29}=1 49. (x−4)225+(y−2)21=1\frac{{\left(x - 4\right)}^{2}}{25}+\frac{{\left(y - 2\right)}^{2}}{1}=1 51. (x+3)216+(y−4)24=1\frac{{\left(x+3\right)}^{2}}{16}+\frac{{\left(y - 4\right)}^{2}}{4}=1 53. x281+y29=1\frac{{x}^{2}}{81}+\frac{{y}^{2}}{9}=1 55. (x+2)24+(y−2)29=1\frac{{\left(x+2\right)}^{2}}{4}+\frac{{\left(y - 2\right)}^{2}}{9}=1 57. Area=12π\text{Area}=12\pi square units 59. Area=25π\text{Area}=2\sqrt{5}\pi square units 61. Area 9π\text{Area }9\pi square units 63. x24h2+y214h2=1\frac{{x}^{2}}{4{h}^{2}}+\frac{{y}^{2}}{\frac{1}{4}{h}^{2}}=1 65. x2400+y2144=1\frac{{x}^{2}}{400}+\frac{{y}^{2}}{144}=1. Distance = 17.32 feet 67. Approximately 51.96 feet

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