10.8.3: Answers to Questions

10.2 Section 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; ${\displaystyle \frac{x^2}{3^2}}+{\displaystyle \frac{y^2}{2^2}}=1$

9. yes; ${\displaystyle \frac{x^2}{{\left({\displaystyle \frac{1}{2}}\right)}^2}}+{\displaystyle \frac{y^2}{{\left({\displaystyle \frac{1}{3}}\right)}^2}}=1$

11. ${\displaystyle \frac{x^2}{2^2}}+{\displaystyle \frac{y^2}{7^2}}=1$; 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. ${\displaystyle \frac{x^2}{{(1)}^2}}+{\displaystyle \frac{y^2}{{\left({\displaystyle \frac{1}{3}}\right)}^2}}=1$; Endpoints of major axis $(1,0)$ and $(-1,0)$. Endpoints of minor axis $\left(0,{\displaystyle \frac{1}{3}}\right),\left(0,-{\displaystyle \frac{1}{3}}\right)$.

Foci at $\left({\displaystyle \frac{2\sqrt{2}}{3}},0\right),\left(-{\displaystyle \frac{2\sqrt{2}}{3}},0\right)$.

15. ${\displaystyle \frac{{(x-2)}^2}{7^2}}+{\displaystyle \frac{{(y-4)}^2}{5^2}}=1$; 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. ${\displaystyle \frac{{(x+5)}^2}{2^2}}+{\displaystyle \frac{{(y-7)}^2}{3^2}}=1$; 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. ${\displaystyle \frac{{(x-1)}^2}{3^2}}+{\displaystyle \frac{{(y-4)}^2}{2^2}}=1$; 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. ${\displaystyle \frac{{(x-3)}^2}{{(3\sqrt{2})}^2}}+{\displaystyle \frac{{(y-5)}^2}{{(\sqrt{2})}^2}}=1$; 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. ${\displaystyle \frac{{(x+5)}^2}{{(5)}^2}}+{\displaystyle \frac{{(y-2)}^2}{{(2)}^2}}=1$; 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. ${\displaystyle \frac{{(x+3)}^2}{{(5)}^2}}+{\displaystyle \frac{{(y+4)}^2}{{(2)}^2}}=1$; 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 $\left({\displaystyle \frac{1}{9}},0\right),\left(-{\displaystyle \frac{1}{9}},0\right),\left(0,{\displaystyle \frac{1}{7}}\right),\left(0,-{\displaystyle \frac{1}{7}}\right)$, Foci $\left(0,{\displaystyle \frac{4\sqrt{2}}{63}}\right),\left(0,-{\displaystyle \frac{4\sqrt{2}}{63}}\right)$

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. ${\displaystyle \frac{x^2}{25}}+{\displaystyle \frac{y^2}{29}}=1$

49. ${\displaystyle \frac{{(x-4)}^2}{25}}+{\displaystyle \frac{{(y-2)}^2}{1}}=1$

51. ${\displaystyle \frac{{(x+3)}^2}{16}}+{\displaystyle \frac{{(y-4)}^2}{4}}=1$

53. ${\displaystyle \frac{x^2}{81}}+{\displaystyle \frac{y^2}{9}}=1$

55. ${\displaystyle \frac{{(x+2)}^2}{4}}+{\displaystyle \frac{{(y-2)}^2}{9}}=1$

57. Area $=12\pi$ square units

59. Area $=2\sqrt{5}\pi$ square units.

61. Area $=9\pi$ square units.

63. ${\displaystyle \frac{x^2}{4h^2}}+{\displaystyle \frac{y^2}{{\displaystyle \frac{1}{4}}{h}^2}}=1$

65. ${\displaystyle \frac{x^2}{400}}+{\displaystyle \frac{y^2}{144}}=1$. Distance $=17.32$ feet

67. Approximately 51.96 feet