Exercises 21.5, page 592

1. $V (0, 6), V (0, - 6),$ $V (0, 6), V (0, - 6),$ ends minor axis (5, 0), $(- 5, 0)$ $(- 5, 0)$ foci $(0, \sqrt{11}), (0, - \sqrt{11})$ $(0, \sqrt{11}), (0, - \sqrt{11})$
3. $\begin{array}{l} V (2, 0), V (- 2, 0), F (\sqrt{3}, 0), \\ F (- \sqrt{3}, 0) \end{array}$ $\begin{array}{l} V (2, 0), V (- 2, 0), F (\sqrt{3}, 0), \\ F (- \sqrt{3}, 0) \end{array}$
5. $\begin{array}{l} V (0, 12), V (0, - 12), F (0, \sqrt{119}), \\ F (0, - \sqrt{119}) \end{array}$ $\begin{array}{l} V (0, 12), V (0, - 12), F (0, \sqrt{119}), \\ F (0, - \sqrt{119}) \end{array}$
7. $\begin{array}{l} V (\frac{5}{2}, 0), V (- \frac{5}{2}, 0), \\ F (\frac{\sqrt{209}}{6}, 0), \\ F (- \frac{\sqrt{209}}{6}, 0) \end{array}$ $\begin{array}{l} V (\frac{5}{2}, 0), V (- \frac{5}{2}, 0), \\ F (\frac{\sqrt{209}}{6}, 0), \\ F (- \frac{\sqrt{209}}{6}, 0) \end{array}$
9. $\begin{array}{l} V (9, 0), V (- 9, 0), \\ F (3 \sqrt{5}, 0), F (- 3 \sqrt{5}, 0) \end{array}$ $\begin{array}{l} V (9, 0), V (- 9, 0), \\ F (3 \sqrt{5}, 0), F (- 3 \sqrt{5}, 0) \end{array}$
11. $\begin{array}{l} V (0, \frac{7}{2} \sqrt{2}), V (0, - \frac{7}{2} \sqrt{2}), \\ F (0, \frac{1}{2} \sqrt{82}), F (0, - \frac{1}{2} \sqrt{82}) \end{array}$ $\begin{array}{l} V (0, \frac{7}{2} \sqrt{2}), V (0, - \frac{7}{2} \sqrt{2}), \\ F (0, \frac{1}{2} \sqrt{82}), F (0, - \frac{1}{2} \sqrt{82}) \end{array}$
13. $\begin{array}{l} V (0, 4), V (0, - 4), \\ F (0, \sqrt{14}), F (0, - \sqrt{14}) \end{array}$ $\begin{array}{l} V (0, 4), V (0, - 4), \\ F (0, \sqrt{14}), F (0, - \sqrt{14}) \end{array}$
15. $\begin{array}{l} V (0.25, 0), V (- 0.25, 0), \\ F (0.23, 0), F (- 0.23, 0) \end{array}$ $\begin{array}{l} V (0.25, 0), V (- 0.25, 0), \\ F (0.23, 0), F (- 0.23, 0) \end{array}$
17. $\frac{x^{2}}{225} + \frac{y^{2}}{144} = 1,$ $\frac{x^{2}}{225} + \frac{y^{2}}{144} = 1,$ or $144 x^{2} + 225 y^{2} = 32,400$ $144 x^{2} + 225 y^{2} = 32,400$
19. $\frac{x^{2}}{225} + \frac{y^{2}}{289} = 1,$ $\frac{x^{2}}{225} + \frac{y^{2}}{289} = 1,$ or $289 x^{2} + 225 y^{2} = 65,025$ $289 x^{2} + 225 y^{2} = 65,025$
21. $\frac{x^{2}}{208} + \frac{y^{2}}{144} = 1,$ $\frac{x^{2}}{208} + \frac{y^{2}}{144} = 1,$ or $144 x^{2} + 208 y^{2} = 29,952$ $144 x^{2} + 208 y^{2} = 29,952$
23. $\frac{x^{2}}{64} + \frac{15 y^{2}}{144} = 1,$ $\frac{x^{2}}{64} + \frac{15 y^{2}}{144} = 1,$ or $3 x^{2} + 20 y^{2} = 192$ $3 x^{2} + 20 y^{2} = 192$
25. $\frac{x^{2}}{5} + \frac{y^{2}}{20} = 1,$ $\frac{x^{2}}{5} + \frac{y^{2}}{20} = 1,$ or $4 x^{2} + y^{2} = 20$ $4 x^{2} + y^{2} = 20$
27. $\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1,$ $\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1,$ or $16 x^{2} + 25 y^{2} = 1600$ $16 x^{2} + 25 y^{2} = 1600$
29. $(1, - 2), (1, 2)$ $(1, - 2), (1, 2)$
31. $16 x^{2} + 25 y^{2} - 32 x - 50 y - 359 = 0$ $16 x^{2} + 25 y^{2} - 32 x - 50 y - 359 = 0$
33.
35.
37. Write equation as $\frac{x^{2}}{1} + \frac{y^{2}}{1 / k} = 1 .$ $\frac{x^{2}}{1} + \frac{y^{2}}{1 / k} = 1 .$ Thus, $\sqrt{\frac{1}{k}} > 1,$ $\sqrt{\frac{1}{k}} > 1,$ or $0 < k < 1 .$ $0 < k < 1 .$
39. $2 x^{2} + 3 y^{2} - 8 x - 4 = 2 x^{2} + 3 {(- y)}^{2} - 8 x - 4$ $2 x^{2} + 3 y^{2} - 8 x - 4 = 2 x^{2} + 3 {(- y)}^{2} - 8 x - 4$
41. $\sin^{2} t + \cos^{2} t = \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$ $\sin^{2} t + \cos^{2} t = \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$
43. $\frac{x^{2}}{256} + \frac{y^{2}}{25} = 1$ $\frac{x^{2}}{256} + \frac{y^{2}}{25} = 1$
45. $i_{1}^{2} + 4 i_{2}^{2} = 32$ $i_{1}^{2} + 4 i_{2}^{2} = 32$
47. 0.44
49. $7 x^{2} + 16 y^{2} = 112$ $7 x^{2} + 16 y^{2} = 112$
51. 27.5m
53. 13 ft
55. ${843 ft}^{3}$ ${843 ft}^{3}$