10.8.3: Answers to Questions
- Page ID
- 152101
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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; \(\dfrac{x^2}{3^2}+\dfrac{y^2}{2^2}=1\)
9. yes; \(\dfrac{x^2}{\left(\dfrac{1}{2}\right)^2}+\dfrac{y^2}{\left(\dfrac{1}{3}\right)^2}=1\)
11. \(\dfrac{x^2}{2^2}+\dfrac{y^2}{7^2}=1\); Endpoints of major axis \((0,7)\) and \((0,-7)\). Endpoints of minor axis \((2,0)\) and \((-2,0)\). Foci at \((0,3 \sqrt{5}),(0,-3 \sqrt{5})\).
13. \(\dfrac{x^2}{(1)^2}+\dfrac{y^2}{\left(\dfrac{1}{3}\right)^2}=1\); Endpoints of major axis \((1,0)\) and \((-1,0)\). Endpoints of minor axis \(\left(0, \dfrac{1}{3}\right),\left(0,-\dfrac{1}{3}\right)\).
Foci at \(\left(\dfrac{2 \sqrt{2}}{3}, 0\right),\left(-\dfrac{2 \sqrt{2}}{3}, 0\right)\).
15. \(\dfrac{(x-2)^2}{7^2}+\dfrac{(y-4)^2}{5^2}=1\); Endpoints of major axis \((9,4),(-5,4)\). Endpoints of minor axis \((2,9),(2,-1)\). Foci at \((2+2 \sqrt{6}, 4)^2,(2-2 \sqrt{6}, 4)\).
17. \(\dfrac{(x+5)^2}{2^2}+\dfrac{(y-7)^2}{3^2}=1\); Endpoints of major axis \((-5,10),(-5,4)\). Endpoints of minor axis \((-3,7),(-7,7)\). Foci at \((-5,7+\sqrt{5}),(-5,7-\sqrt{5})\).
19. \(\dfrac{(x-1)^2}{3^2}+\dfrac{(y-4)^2}{2^2}=1\); Endpoints of major axis \((4,4),(-2,4)\). Endpoints of minor axis \((1,6),(1,2)\). Foci at \((1+\sqrt[3]{5}, 4),(1-\sqrt{5}, 4)\).
21. \(\dfrac{(x-3)^2}{(3 \sqrt{2})^2}+\dfrac{(y-5)^2}{(\sqrt{2})^2}=1\); Endpoints of major axis \((3+3 \sqrt{2}, 5),(3-3 \sqrt{2}, 5)\). Endpoints of minor axis \((3,5+\sqrt{2}),(3,5-\sqrt{2})\). Foci at \((7,5),(-1,5)\).
23. \(\dfrac{(x+5)^2}{(5)^2}+\dfrac{(y-2)^2}{(2)^2}=1\); Endpoints of major axis \((0,2),(-10,2)\). Endpoints of minor axis \((-5,4),(-5,0)\). Foci at \((-5+\sqrt{21}, 2),(-5-\sqrt{21}, 2)\).
25. \(\dfrac{(x+3)^2}{(5)^2}+\dfrac{(y+4)^2}{(2)^2}=1\); Endpoints of major axis \((2,-4),(-8,-4)\). Endpoints of minor axis \((-3,-2),(-3,-6)\). Foci at \((-3+\sqrt{21},-4),(-3-\sqrt{21},-4)\).
27. Foci \((-3,-1+\sqrt{11}),(-3,-1-\sqrt{11})\)
29. Focus \((0,0)\)
31. Foci \((-10,30),(-10,-30)\)
33. Center \((0,0)\), Vertices \((4,0),(-4,0),(0,3),(0,-3)\), Foci \((\sqrt{7}, 0),(-\sqrt{7}, 0)\)
35. Center \((0,0)\), Vertices \(\left(\dfrac{1}{9}, 0\right),\left(-\dfrac{1}{9}, 0\right),\left(0, \dfrac{1}{7}\right),\left(0,-\dfrac{1}{7}\right)\), Foci \(\left(0, \dfrac{4 \sqrt{2}}{63}\right),\left(0,-\dfrac{4 \sqrt{2}}{63}\right)\)
37. Center \((-3,3)\), Vertices \((0,3),(-6,3),(-3,0),(-3,6)\), Focus \((-3,3)\)
Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.
39. Center \((1,1)\), Vertices \((5,1),(-3,1),(1,3),(1,-1)\), Foci \((1+2 \sqrt{3}, 1),(1-2 \sqrt{3}, 1)\)
41. Center \((-4,5)\), Vertices \((-2,5),(-6,4),(-4,6),(-4,4)\), Foci \((-4+\sqrt{3}, 5),(-4-\sqrt{3}, 5)\)
43. Center \((-2,1)\), Vertices \((0,1),(-4,1),(-2,5),(-2,-3)\), Foci \((-2,1+2 \sqrt{3}),(-2,1-2 \sqrt{3})\)
45. Center \((-2,-2)\), Vertices \((0,-2),(-4,-2),(-2,0),(-2,-4)\), Focus \((-2,-2)\)
47. \(\dfrac{x^2}{25}+\dfrac{y^2}{29}=1\)
49. \(\dfrac{(x-4)^2}{25}+\dfrac{(y-2)^2}{1}=1\)
51. \(\dfrac{(x+3)^2}{16}+\dfrac{(y-4)^2}{4}=1\)
53. \(\dfrac{x^2}{81}+\dfrac{y^2}{9}=1\)
55. \(\dfrac{(x+2)^2}{4}+\dfrac{(y-2)^2}{9}=1\)
57. Area \(=12 \pi\) square units
59. Area \(=2 \sqrt{5} \pi\) square units.
61. Area \(=9 \pi\) square units.
63. \(\dfrac{x^2}{4 h^2}+\dfrac{y^2}{\dfrac{1}{4} h^2}=1\)
65. \(\dfrac{x^2}{400}+\dfrac{y^2}{144}=1\). Distance \(=17.32\) feet
67. Approximately 51.96 feet