Solutions to Odd-Numbered Exercises

1. A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

3. The foci must lie on the transverse axis and be in the interior of the hyperbola.

5. The center must be the midpoint of the line segment joining the foci.

7. yes $\frac{{x}^{2}}{{6}^{2}}-\frac{{y}^{2}}{{3}^{2}}=1$

9. yes $\frac{{x}^{2}}{{4}^{2}}-\frac{{y}^{2}}{{5}^{2}}=1$

11. $\frac{{x}^{2}}{{5}^{2}}-\frac{{y}^{2}}{{6}^{2}}=1$; vertices: $\left(5,0\right),\left(-5,0\right)$; foci: $\left(\sqrt{61},0\right),\left(-\sqrt{61},0\right)$; asymptotes: $y=\frac{6}{5}x,y=-\frac{6}{5}x$

13. $\frac{{y}^{2}}{{2}^{2}}-\frac{{x}^{2}}{{9}^{2}}=1$; vertices: $\left(0,2\right),\left(0,-2\right)$; foci: $\left(0,\sqrt{85}\right),\left(0,-\sqrt{85}\right)$; asymptotes: $y=\frac{2}{9}x,y=-\frac{2}{9}x$

15. $\frac{{\left(x - 1\right)}^{2}}{{3}^{2}}-\frac{{\left(y - 2\right)}^{2}}{{4}^{2}}=1$; vertices: $\left(4,2\right),\left(-2,2\right)$; foci: $\left(6,2\right),\left(-4,2\right)$; asymptotes: $y=\frac{4}{3}\left(x - 1\right)+2,y=-\frac{4}{3}\left(x - 1\right)+2$

17. $\frac{{\left(x - 2\right)}^{2}}{{7}^{2}}-\frac{{\left(y+7\right)}^{2}}{{7}^{2}}=1$; vertices: $\left(9,-7\right),\left(-5,-7\right)$; foci: $\left(2+7\sqrt{2},-7\right),\left(2 - 7\sqrt{2},-7\right)$; asymptotes: $y=x - 9,y=-x - 5$

19. $\frac{{\left(x+3\right)}^{2}}{{3}^{2}}-\frac{{\left(y - 3\right)}^{2}}{{3}^{2}}=1$; vertices: $\left(0,3\right),\left(-6,3\right)$; foci: $\left(-3+3\sqrt{2},1\right),\left(-3 - 3\sqrt{2},1\right)$; asymptotes: $y=x+6,y=-x$

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

23. $\frac{{\left(y+5\right)}^{2}}{{7}^{2}}-\frac{{\left(x+1\right)}^{2}}{{70}^{2}}=1$; vertices: $\left(-1,2\right),\left(-1,-12\right)$; foci: $\left(-1,-5+7\sqrt{101}\right),\left(-1,-5 - 7\sqrt{101}\right)$; asymptotes: $y=\frac{1}{10}\left(x+1\right)-5,y=-\frac{1}{10}\left(x+1\right)-5$

25. $\frac{{\left(x+3\right)}^{2}}{{5}^{2}}-\frac{{\left(y - 4\right)}^{2}}{{2}^{2}}=1$; vertices: $\left(2,4\right),\left(-8,4\right)$; foci: $\left(-3+\sqrt{29},4\right),\left(-3-\sqrt{29},4\right)$; asymptotes: $y=\frac{2}{5}\left(x+3\right)+4,y=-\frac{2}{5}\left(x+3\right)+4$

27. $y=\frac{2}{5}\left(x - 3\right)-4,y=-\frac{2}{5}\left(x - 3\right)-4$

29. $y=\frac{3}{4}\left(x - 1\right)+1,y=-\frac{3}{4}\left(x - 1\right)+1$

31.

33.

35.

37.

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45. $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{16}=1$

47. $\frac{{\left(x - 6\right)}^{2}}{25}-\frac{{\left(y - 1\right)}^{2}}{11}=1$

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

51. $\frac{{y}^{2}}{16}-\frac{{x}^{2}}{25}=1$

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

55. $\frac{{\left(x+3\right)}^{2}}{25}-\frac{{\left(y+3\right)}^{2}}{25}=1$

57. $y\left(x\right)=3\sqrt{{x}^{2}+1},y\left(x\right)=-3\sqrt{{x}^{2}+1}$

59. $y\left(x\right)=1+2\sqrt{{x}^{2}+4x+5},y\left(x\right)=1 - 2\sqrt{{x}^{2}+4x+5}$

61. $\frac{{x}^{2}}{25}-\frac{{y}^{2}}{25}=1$

63. $\frac{{x}^{2}}{100}-\frac{{y}^{2}}{25}=1$

65. $\frac{{x}^{2}}{400}-\frac{{y}^{2}}{225}=1$

67. $\frac{{\left(x - 1\right)}^{2}}{0.25}-\frac{{y}^{2}}{0.75}=1$

69. $\frac{{\left(x - 3\right)}^{2}}{4}-\frac{{y}^{2}}{5}=1$