Solutions to Try Its

1. ${x}^{2}+\frac{{y}^{2}}{16}=1$

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

3. center: $\left(0,0\right)$; vertices: $\left(\pm 6,0\right)$; co-vertices: $\left(0,\pm 2\right)$; foci: $\left(\pm 4\sqrt{2},0\right)$

4. Standard form: $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{49}=1$; center: $\left(0,0\right)$; vertices: $\left(0,\pm 7\right)$; co-vertices: $\left(\pm 4,0\right)$; foci: $\left(0,\pm \sqrt{33}\right)$

5. Center: $\left(4,2\right)$; vertices: $\left(-2,2\right)$ and $\left(10,2\right)$; co-vertices: $\left(4,2 - 2\sqrt{5}\right)$ and $\left(4,2+2\sqrt{5}\right)$; foci: $\left(0,2\right)$ and $\left(8,2\right)$

6. $\frac{{\left(x - 3\right)}^{2}}{4}+\frac{{\left(y+1\right)}^{2}}{16}=1$; center: $\left(3,-1\right)$; vertices: $\left(3,-\text{5}\right)$ and $\left(3,\text{3}\right)$; co-vertices: $\left(1,-1\right)$ and $\left(5,-1\right)$; foci: $\left(3,-\text{1}-2\sqrt{3}\right)$ and $\left(3,-\text{1+}2\sqrt{3}\right)$

7. a. $\frac{{x}^{2}}{57,600}+\frac{{y}^{2}}{25,600}=1$
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; $\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1$

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

11. $\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{7}^{2}}=1$; Endpoints of major axis $\left(0,7\right)$ and $\left(0,-7\right)$. Endpoints of minor axis $\left(2,0\right)$ and $\left(-2,0\right)$. Foci at $\left(0,3\sqrt{5}\right),\left(0,-3\sqrt{5}\right)$.

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

15. $\frac{{\left(x - 2\right)}^{2}}{{7}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{5}^{2}}=1$; Endpoints of major axis $\left(9,4\right),\left(-5,4\right)$. Endpoints of minor axis $\left(2,9\right),\left(2,-1\right)$. Foci at $\left(2+2\sqrt{6},4\right),\left(2 - 2\sqrt{6},4\right)$.

17. $\frac{{\left(x+5\right)}^{2}}{{2}^{2}}+\frac{{\left(y - 7\right)}^{2}}{{3}^{2}}=1$; Endpoints of major axis $\left(-5,10\right),\left(-5,4\right)$. Endpoints of minor axis $\left(-3,7\right),\left(-7,7\right)$. Foci at $\left(-5,7+\sqrt{5}\right),\left(-5,7-\sqrt{5}\right)$.

19. $\frac{{\left(x - 1\right)}^{2}}{{3}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{2}^{2}}=1$; Endpoints of major axis $\left(4,4\right),\left(-2,4\right)$. Endpoints of minor axis $\left(1,6\right),\left(1,2\right)$. Foci at $\left(1+\sqrt{5},4\right),\left(1-\sqrt{5},4\right)$.

21. $\frac{{\left(x - 3\right)}^{2}}{{\left(3\sqrt{2}\right)}^{2}}+\frac{{\left(y - 5\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}=1$; Endpoints of major axis $\left(3+3\sqrt{2},5\right),\left(3 - 3\sqrt{2},5\right)$. Endpoints of minor axis $\left(3,5+\sqrt{2}\right),\left(3,5-\sqrt{2}\right)$. Foci at $\left(7,5\right),\left(-1,5\right)$.

23. $\frac{{\left(x+5\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y - 2\right)}^{2}}{{\left(2\right)}^{2}}=1$; Endpoints of major axis $\left(0,2\right),\left(-10,2\right)$. Endpoints of minor axis $\left(-5,4\right),\left(-5,0\right)$. Foci at $\left(-5+\sqrt{21},2\right),\left(-5-\sqrt{21},2\right)$.

25. $\frac{{\left(x+3\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y+4\right)}^{2}}{{\left(2\right)}^{2}}=1$; Endpoints of major axis $\left(2,-4\right),\left(-8,-4\right)$. Endpoints of minor axis $\left(-3,-2\right),\left(-3,-6\right)$. Foci at $\left(-3+\sqrt{21},-4\right),\left(-3-\sqrt{21},-4\right)$.

27. Foci $\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right)$

29. Focus $\left(0,0\right)$

31. Foci $\left(-10,30\right),\left(-10,-30\right)$

33. Center $\left(0,0\right)$, Vertices $\left(4,0\right),\left(-4,0\right),\left(0,3\right),\left(0,-3\right)$, Foci $\left(\sqrt{7},0\right),\left(-\sqrt{7},0\right)$

35. Center $\left(0,0\right)$, Vertices $\left(\frac{1}{9},0\right),\left(-\frac{1}{9},0\right),\left(0,\frac{1}{7}\right),\left(0,-\frac{1}{7}\right)$, Foci $\left(0,\frac{4\sqrt{2}}{63}\right),\left(0,-\frac{4\sqrt{2}}{63}\right)$

37. Center $\left(-3,3\right)$, Vertices $\left(0,3\right),\left(-6,3\right),\left(-3,0\right),\left(-3,6\right)$, Focus $\left(-3,3\right)$
Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

39. Center $\left(1,1\right)$, Vertices $\left(5,1\right),\left(-3,1\right),\left(1,3\right),\left(1,-1\right)$, Foci $\left(1,1+4\sqrt{3}\right),\left(1,1 - 4\sqrt{3}\right)$

41. Center $\left(-4,5\right)$, Vertices $\left(-2,5\right),\left(-6,4\right),\left(-4,6\right),\left(-4,4\right)$, Foci $\left(-4+\sqrt{3},5\right),\left(-4-\sqrt{3},5\right)$

43. Center $\left(-2,1\right)$, Vertices $\left(0,1\right),\left(-4,1\right),\left(-2,5\right),\left(-2,-3\right)$, Foci $\left(-2,1+2\sqrt{3}\right),\left(-2,1 - 2\sqrt{3}\right)$

45. Center $\left(-2,-2\right)$, Vertices $\left(0,-2\right),\left(-4,-2\right),\left(-2,0\right),\left(-2,-4\right)$, Focus $\left(-2,-2\right)$

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

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

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

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

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

57. $\text{Area}=12\pi$ square units

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

61. $\text{Area }9\pi$ square units

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

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

67. Approximately 51.96 feet