Interactive Algebra & Trigonometry - A Self Guided Study Companion

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`1%`	dexvert	JuggleKrazy Tutorial `(other/juggleKrazyTutorial)`	ext	Unsupported
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|00000df0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00000e10| 20 20 20 20 20 20 20 11 | 34 44 32 0d 0b 00 20 20 |       .|4D2...  |
|00000e20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000e30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000e40| 20 11 33 62 20 11 31 3d | 20 11 34 53 20 11 31 36 | .3b .1=| .4S .16|
|00000e50| 13 0d 0a 00 0d 0b 00 54 | 68 65 72 65 66 6f 72 65 |.......T|herefore|
|00000e60| 2c 20 6f 75 72 20 65 71 | 75 61 74 69 6f 6e 20 69 |, our eq|uation i|
|00000e70| 73 20 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |s ...   |        |
|00000e80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000e90| 20 20 11 32 32 20 20 20 | 20 32 0d 0b 00 20 20 20 |  .22   | 2...   |
|00000ea0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000eb0| 20 20 20 20 20 20 20 20 | 20 11 33 79 20 20 20 20 |        | .3y    |
|00000ec0| 78 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |x...    |        |
|00000ed0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000ee0| 11 34 32 32 20 11 31 2d | 20 11 34 32 32 20 11 31 |.422 .1-| .422 .1|
|00000ef0| 3d 20 31 2e 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |= 1.... |        |
|00000f00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000f10| 20 20 20 31 35 20 20 20 | 36 0d 0a 00 53 65 63 74 |   15   |6...Sect|
|00000f20| 69 6f 6e 20 34 2e 34 20 | 20 43 6f 6e 69 63 73 0d |ion 4.4 | Conics.|
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|00000f50| 20 77 69 74 68 20 69 74 | 73 20 76 65 72 74 65 78 | with it|s vertex|
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|00000f80| 2d 31 29 2e 0d 0a 00 0d | 0b 00 13 12 31 53 4f 4c |-1).....|....1SOL|
|00000f90| 55 54 49 4f 4e 12 30 0d | 0a 00 0d 0b 00 53 69 6e |UTION.0.|.....Sin|
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|00000fc0| 65 72 74 65 78 2c 20 28 | 30 2c 20 30 29 2c 20 61 |ertex, (|0, 0), a|
|00000fd0| 6e 64 20 74 68 65 20 66 | 6f 63 75 73 2c 20 28 30 |nd the f|ocus, (0|
|00000fe0| 2c 20 2d 31 29 2c 0d 0a | 00 20 20 20 20 20 20 20 |, -1),..|.       |
|00000ff0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001000| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001010| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001020| 20 20 20 20 20 11 32 32 | 0d 0b 00 11 31 77 65 20 |     .22|....1we |
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|000010d0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |.       |      .2|
|000010e0| 32 0d 0b 00 11 31 65 71 | 75 61 74 69 6f 6e 20 69 |2....1eq|uation i|
|000010f0| 73 20 11 33 78 20 20 11 | 31 3d 20 34 28 2d 31 29 |s .3x  .|1= 4(-1)|
|00001100| 11 33 79 0d 0a 00 0d 0b | 00 20 20 20 20 20 20 20 |.3y.....|.       |
|00001110| 20 20 20 20 20 20 20 20 | 11 31 3d 20 2d 34 11 33 |        |.1= -4.3|
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|00001130| 2e 34 20 20 43 6f 6e 69 | 63 73 0d 0b 00 46 69 6e |.4  Coni|cs...Fin|
|00001140| 64 20 61 6e 20 65 71 75 | 61 74 69 6f 6e 20 6f 66 |d an equ|ation of|
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|00001180| 72 65 63 74 72 69 78 0d | 0a 00 11 33 79 20 11 31 |rectrix.|...3y .1|
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|000011d0| 20 66 6f 72 6d 20 11 33 | 79 20 11 31 3d 20 2d 11 | form .3|y .1= -.|
|000011e0| 33 70 20 11 31 77 65 20 | 63 61 6e 20 69 6d 6d 65 |3p .1we |can imme|
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|00001220| 61 6e 64 20 74 68 61 74 | 20 2d 11 33 70 20 11 31 |and that| -.3p .1|
|00001230| 3d 20 32 20 6f 72 20 11 | 33 70 20 11 31 3d 20 2d |= 2 or .|3p .1= -|
|00001240| 32 2e 13 0d 0a 00 0d 0b | 00 46 72 6f 6d 20 74 68 |2.......|.From th|
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|00001260| 74 68 65 20 66 6f 63 75 | 73 20 69 73 20 61 74 20 |the focu|s is at |
|00001270| 28 30 2c 20 2d 32 29 20 | 61 6e 64 20 77 65 20 63 |(0, -2) |and we c|
|00001280| 6f 6e 73 69 64 65 72 20 | 74 68 65 20 66 6f 72 6d |onsider |the form|
|00001290| 0d 0a 00 20 11 32 32 0d | 0b 00 11 33 78 20 20 11 |... .22.|...3x  .|
|000012a0| 31 3d 20 34 11 33 70 79 | 11 31 2e 13 0d 0a 00 20 |1= 4.3py|.1..... |
|000012b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000012c0| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
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|00001300| 0a 00 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |.....   |        |
|00001310| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001320| 20 20 20 11 31 3d 20 2d | 38 11 33 79 11 31 2e 0d |   .1= -|8.3y.1..|
|00001330| 0a 00 53 65 63 74 69 6f | 6e 20 34 2e 34 20 20 43 |..Sectio|n 4.4  C|
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|00001360| 72 61 62 6f 6c 61 20 77 | 69 74 68 20 69 74 73 20 |rabola w|ith its |
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|00001390| 72 69 7a 6f 6e 74 61 6c | 20 61 78 69 73 20 61 6e |rizontal| axis an|
|000013a0| 64 20 70 61 73 73 65 73 | 20 74 68 72 6f 75 67 68 |d passes| through|
|000013b0| 20 74 68 65 20 70 6f 69 | 6e 74 20 28 2d 31 2c 20 | the poi|nt (-1, |
|000013c0| 32 29 2e 0d 0a 00 0d 0b | 00 13 12 31 53 4f 4c 55 |2)......|...1SOLU|
|000013d0| 54 49 4f 4e 12 30 0d 0a | 00 20 20 20 20 20 20 20 |TION.0..|.       |
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|000013f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001400| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001410| 32 32 0d 0b 00 11 31 53 | 69 6e 63 65 20 77 65 20 |22....1S|ince we |
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|00001450| 20 11 31 3d 20 34 11 33 | 70 78 11 31 2e 13 0d 0a | .1= 4.3|px.1....|
|00001460| 00 0d 0b 00 4b 6e 6f 77 | 69 6e 67 20 74 68 61 74 |....Know|ing that|
|00001470| 20 74 68 65 20 67 72 61 | 70 68 20 70 61 73 73 65 | the gra|ph passe|
|00001480| 73 20 74 68 72 6f 75 67 | 68 20 74 68 65 20 70 6f |s throug|h the po|
|00001490| 69 6e 74 20 28 2d 31 2c | 20 32 29 20 77 65 20 63 |int (-1,| 2) we c|
|000014a0| 61 6e 20 70 6c 75 67 20 | 74 68 65 73 65 0d 0a 00 |an plug |these...|
|000014b0| 76 61 6c 75 65 73 20 69 | 6e 74 6f 20 74 68 65 20 |values i|nto the |
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|000014e0| 20 20 20 20 20 20 11 32 | 32 0d 0b 00 20 20 20 20 |      .2|2...    |
|000014f0| 20 11 31 32 20 20 3d 20 | 34 11 33 70 11 31 28 2d | .12  = |4.3p.1(-|
|00001500| 31 29 0d 0a 00 20 20 20 | 20 20 34 20 3d 20 2d 34 |1)...   |  4 = -4|
|00001510| 11 33 70 0d 0a 00 20 20 | 20 20 11 31 2d 31 20 3d |.3p...  |  .1-1 =|
|00001520| 20 11 33 70 11 31 13 0d | 0a 00 0d 0b 00 54 68 65 | .3p.1..|.....The|
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|00001540| 74 68 65 20 66 6f 63 75 | 73 20 69 73 20 61 74 20 |the focu|s is at |
|00001550| 28 11 33 70 11 31 2c 20 | 30 29 20 3d 20 28 2d 31 |(.3p.1, |0) = (-1|
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|00001580| 20 11 32 32 0d 0b 00 20 | 20 20 20 20 11 33 79 20 | .22... |    .3y |
|00001590| 20 11 31 3d 20 2d 34 11 | 33 78 11 31 2e 0d 0a 00 | .1= -4.|3x.1....|
|000015a0| 53 65 63 74 69 6f 6e 20 | 34 2e 34 20 20 43 6f 6e |Section |4.4  Con|
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|00001610| 20 20 20 20 11 32 32 20 | 20 20 20 20 32 0d 0b 00 |    .22 |    2...|
|00001620| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001630| 20 20 20 11 31 31 32 11 | 33 79 20 20 11 31 2b 20 |   .112.|3y  .1+ |
|00001640| 33 11 33 78 20 20 11 31 | 3d 20 31 32 0d 0a 00 0d |3.3x  .1|= 12....|
|00001650| 0b 00 13 12 31 53 4f 4c | 55 54 49 4f 4e 12 30 0d |....1SOL|UTION.0.|
|00001660| 0a 00 0d 0b 00 57 65 20 | 62 65 67 69 6e 20 62 79 |.....We |begin by|
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|000016a0| 20 20 11 32 32 20 20 20 | 20 20 32 0d 0b 00 20 20 |  .22   |  2...  |
|000016b0| 20 20 20 11 31 31 32 11 | 33 79 20 20 11 31 2b 20 |   .112.|3y  .1+ |
|000016c0| 33 11 33 78 20 20 11 31 | 3d 20 31 32 20 20 20 20 |3.3x  .1|= 12    |
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|00001700| 00 20 20 20 20 20 20 20 | 20 11 32 32 20 20 20 20 |.       | .22    |
|00001710| 20 32 0d 0b 00 20 20 20 | 20 20 11 31 31 32 11 33 | 2...   |  .112.3|
|00001720| 79 20 20 20 20 11 31 33 | 11 33 78 20 20 20 20 11 |y    .13|.3x    .|
|00001730| 31 31 32 0d 0b 00 20 20 | 20 20 20 11 34 32 32 32 |112...  |   .4222|
|00001740| 32 20 11 31 2b 20 11 34 | 32 32 32 20 11 31 3d 20 |2 .1+ .4|222 .1= |
|00001750| 11 34 32 32 0d 0b 00 20 | 20 20 20 20 20 11 31 31 |.422... |     .11|
|00001760| 32 20 20 20 20 31 32 20 | 20 20 20 31 32 13 0d 0a |2    12 |   12...|
|00001770| 00 20 20 20 20 20 20 20 | 20 20 11 32 32 20 20 20 |.       |  .22   |
|00001780| 20 32 0d 0b 00 20 20 20 | 20 20 20 20 20 11 33 79 | 2...   |     .3y|
|00001790| 20 20 20 20 78 0d 0b 00 | 20 20 20 20 20 20 20 20 |    x...|        |
|000017a0| 11 34 32 32 20 11 31 2b | 20 11 34 32 32 20 11 31 |.422 .1+| .422 .1|
|000017b0| 3d 20 31 0d 0b 00 20 20 | 20 20 20 20 20 20 31 20 |= 1...  |      1 |
|000017c0| 20 20 20 34 20 20 20 20 | 20 13 0d 0a 00 20 20 20 |   4    | ....   |
|000017d0| 20 20 20 20 20 20 11 32 | 32 20 20 20 20 32 0d 0b |      .2|2    2..|
|000017e0| 00 20 20 20 20 20 20 20 | 20 11 33 79 20 20 20 20 |.       | .3y    |
|000017f0| 78 0d 0b 00 20 20 20 20 | 20 20 20 20 11 34 32 32 |x...    |    .422|
|00001800| 20 11 31 2b 20 11 34 32 | 32 20 11 31 3d 20 31 0d | .1+ .42|2 .1= 1.|
|00001810| 0b 00 20 20 20 20 20 20 | 20 20 20 11 32 32 20 20 |..      |   .22  |
|00001820| 20 20 32 0d 0b 00 20 20 | 20 20 20 20 20 20 11 31 |  2...  |      .1|
|00001830| 31 20 20 20 20 32 13 0d | 0a 00 0d 0a 00 0d 0b 00 |1    2..|........|
|00001840| 54 68 69 73 20 69 73 20 | 74 68 65 20 73 74 61 6e |This is |the stan|
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|000018b0| 13 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|000018c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00001a40| 6d 6f 6f 74 68 20 63 75 | 72 76 65 2e 0d 0a 00 0d |mooth cu|rve.....|
|00001a50| 0a 00 0d 0a 00 0d 0a 00 | 14 6b 2d 35 2d 33 2d 35 |........|.k-5-3-5|
|00001a60| 2e 68 69 14 32 30 14 31 | 38 14 31 38 14 38 14 0d |.hi.20.1|8.18.8..|
|00001a70| 0a 00 0d 0a 00 0d 0a 00 | 53 65 63 74 69 6f 6e 20 |........|Section |
|00001a80| 34 2e 34 20 20 43 6f 6e | 69 63 73 0d 0b 00 46 69 |4.4  Con|ics...Fi|
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|00001ad0| 73 20 61 74 0d 0a 00 28 | 11 34 2b 11 31 33 2c 20 |s at...(|.4+.13, |
|00001ae0| 30 29 2c 20 61 6e 64 20 | 66 6f 63 69 20 61 74 20 |0), and |foci at |
|00001af0| 28 11 34 2b 11 31 31 2c | 20 30 29 2e 0d 0a 00 0d |(.4+.11,| 0).....|
|00001b00| 0b 00 13 12 31 53 4f 4c | 55 54 49 4f 4e 12 30 0d |....1SOL|UTION.0.|
|00001b10| 0a 00 0d 0b 00 53 69 6e | 63 65 20 74 68 65 20 76 |.....Sin|ce the v|
|00001b20| 65 72 74 69 63 65 73 20 | 6f 63 63 75 72 20 61 74 |ertices |occur at|
|00001b30| 20 28 33 2c 20 30 29 20 | 61 6e 64 20 28 2d 33 2c | (3, 0) |and (-3,|
|00001b40| 20 30 29 20 77 65 20 6b | 6e 6f 77 20 74 68 61 74 | 0) we k|now that|
|00001b50| 20 74 68 65 20 6d 61 6a | 6f 72 20 61 78 69 73 0d | the maj|or axis.|
|00001b60| 0a 00 69 73 20 68 6f 72 | 69 7a 6f 6e 74 61 6c 20 |..is hor|izontal |
|00001b70| 61 6e 64 20 11 33 61 20 | 11 31 3d 20 33 2e 13 0d |and .3a |.1= 3...|
|00001b80| 0a 00 0d 0b 00 53 69 6e | 63 65 20 74 68 65 20 66 |.....Sin|ce the f|
|00001b90| 6f 63 69 20 6f 63 63 75 | 72 20 61 74 20 28 31 2c |oci occu|r at (1,|
|00001ba0| 20 30 29 20 61 6e 64 20 | 28 2d 31 2c 20 30 29 20 | 0) and |(-1, 0) |
|00001bb0| 77 65 20 6b 6e 6f 77 20 | 74 68 61 74 20 11 33 63 |we know |that .3c|
|00001bc0| 20 11 31 3d 20 31 2e 13 | 0d 0a 00 20 20 20 20 20 | .1= 1..|...     |
|00001bd0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001be0| 32 32 20 20 20 20 32 20 | 20 20 20 32 0d 0b 00 11 |22    2 |   2....|
|00001bf0| 31 55 73 69 6e 67 20 74 | 68 65 20 65 71 75 61 74 |1Using t|he equat|
|00001c00| 69 6f 6e 20 11 33 63 20 | 20 11 31 3d 20 11 33 61 |ion .3c | .1= .3a|
|00001c10| 20 20 11 31 2d 20 11 33 | 62 20 20 11 31 77 65 20 |  .1- .3|b  .1we |
|00001c20| 63 61 6e 20 6e 6f 77 20 | 73 6f 6c 76 65 20 66 6f |can now |solve fo|
|00001c30| 72 20 11 33 62 11 31 2e | 0d 0a 00 20 20 20 20 20 |r .3b.1.|...     |
|00001c40| 20 11 32 32 20 20 20 20 | 32 20 20 20 20 32 0d 0b | .22    |2    2..|
|00001c50| 00 20 20 20 20 20 11 31 | 31 20 20 3d 20 33 20 20 |.     .1|1  = 3  |
|00001c60| 2d 20 11 33 62 0d 0a 00 | 20 20 20 20 20 20 20 20 |- .3b...|        |
|00001c70| 20 20 20 11 34 44 32 0d | 0b 00 20 20 20 20 20 20 |   .4D2.|..      |
|00001c80| 11 33 62 20 11 31 3d 20 | 11 34 53 20 11 31 38 13 |.3b .1= |.4S .18.|
|00001c90| 0d 0a 00 0d 0b 00 54 68 | 65 72 65 66 6f 72 65 2c |......Th|erefore,|
|00001ca0| 20 74 68 65 20 65 71 75 | 61 74 69 6f 6e 20 69 73 | the equ|ation is|
|00001cb0| 0d 0a 00 20 20 20 20 20 | 20 11 32 32 20 20 20 20 |...     | .22    |
|00001cc0| 32 0d 0b 00 20 20 20 20 | 20 11 33 78 20 20 20 20 |2...    | .3x    |
|00001cd0| 79 0d 0b 00 20 20 20 20 | 20 11 34 32 32 20 11 31 |y...    | .422 .1|
|00001ce0| 2b 20 11 34 32 32 20 11 | 31 3d 20 31 2e 0d 0b 00 |+ .422 .|1= 1....|
|00001cf0| 20 20 20 20 20 39 20 20 | 20 20 38 0d 0a 00 53 65 |     9  |  8...Se|
|00001d00| 63 74 69 6f 6e 20 34 2e | 34 20 20 43 6f 6e 69 63 |ction 4.|4  Conic|
|00001d10| 73 0d 0b 00 46 69 6e 64 | 20 61 6e 20 65 71 75 61 |s...Find| an equa|
|00001d20| 74 69 6f 6e 20 6f 66 20 | 74 68 65 20 65 6c 6c 69 |tion of |the elli|
|00001d30| 70 73 65 20 77 69 74 68 | 20 63 65 6e 74 65 72 20 |pse with| center |
|00001d40| 61 74 20 74 68 65 20 6f | 72 69 67 69 6e 2c 20 76 |at the o|rigin, v|
|00001d50| 65 72 74 69 63 65 73 20 | 0d 0a 00 20 20 20 20 20 |ertices |...     |
|00001d60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001d70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001d80| 20 11 34 28 20 20 20 20 | 20 44 32 29 0d 0b 00 11 | .4(    | D2)....|
|00001d90| 31 28 30 2c 20 11 34 2b | 11 31 36 29 2c 20 61 6e |1(0, .4+|.16), an|
|00001da0| 64 20 70 61 73 73 65 73 | 20 74 68 72 6f 75 67 68 |d passes| through|
|00001db0| 20 74 68 65 20 70 6f 69 | 6e 74 20 11 34 21 11 31 | the poi|nt .4!.1|
|00001dc0| 31 2c 20 34 11 34 53 20 | 11 31 32 11 34 21 11 31 |1, 4.4S |.12.4!.1|
|00001dd0| 2e 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|00001de0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001df0| 20 20 20 20 20 20 20 20 | 20 20 11 34 39 20 20 20 |        |  .49   |
|00001e00| 20 20 20 20 30 0d 0a 00 | 0d 0b 00 11 31 13 12 31 |    0...|....1..1|
|00001e10| 53 4f 4c 55 54 49 4f 4e | 12 30 0d 0a 00 0d 0b 00 |SOLUTION|.0......|
|00001e20| 53 69 6e 63 65 20 74 68 | 65 20 76 65 72 74 69 63 |Since th|e vertic|
|00001e30| 65 73 20 6f 63 63 75 72 | 20 61 74 20 28 30 2c 20 |es occur| at (0, |
|00001e40| 36 29 20 61 6e 64 20 28 | 30 2c 20 2d 36 29 20 77 |6) and (|0, -6) w|
|00001e50| 65 20 6b 6e 6f 77 20 74 | 68 61 74 20 74 68 65 20 |e know t|hat the |
|00001e60| 6d 61 6a 6f 72 20 61 78 | 69 73 0d 0a 00 69 73 20 |major ax|is...is |
|00001e70| 76 65 72 74 69 63 61 6c | 20 61 6e 64 20 11 33 61 |vertical| and .3a|
|00001e80| 20 11 31 3d 20 36 2e 13 | 0d 0a 00 0d 0b 00 54 68 | .1= 6..|......Th|
|00001e90| 65 72 65 66 6f 72 65 20 | 77 65 20 68 61 76 65 20 |erefore |we have |
|00001ea0| 74 68 65 20 66 6f 72 6d | 0d 0a 00 20 20 20 20 20 |the form|...     |
|00001eb0| 20 11 32 32 20 20 20 20 | 32 20 20 20 20 20 20 20 | .22    |2       |
|00001ec0| 20 20 20 20 20 20 20 20 | 20 20 32 20 20 20 20 32 |        |  2    2|
|00001ed0| 0d 0b 00 20 20 20 20 20 | 11 33 78 20 20 20 20 79 |...     |.3x    y|
|00001ee0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001ef0| 20 78 20 20 20 20 79 0d | 0b 00 20 20 20 20 20 11 | x    y.|..     .|
|00001f00| 34 32 32 20 11 31 2b 20 | 11 34 32 32 20 11 31 3d |422 .1+ |.422 .1=|
|00001f10| 20 31 20 20 20 20 20 6f | 72 20 20 20 20 20 11 34 | 1     o|r     .4|
|00001f20| 32 32 20 11 31 2b 20 11 | 34 32 32 20 11 31 3d 20 |22 .1+ .|422 .1= |
|00001f30| 31 2e 0d 0b 00 20 20 20 | 20 20 20 11 32 32 20 20 |1....   |   .22  |
|00001f40| 20 20 32 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |  2     |        |
|00001f50| 20 20 20 20 32 0d 0b 00 | 20 20 20 20 20 11 33 62 |    2...|     .3b|
|00001f60| 20 20 20 20 61 20 20 20 | 20 20 20 20 20 20 20 20 |    a   |        |
|00001f70| 20 20 20 20 20 20 62 20 | 20 20 20 11 31 33 36 20 |      b |   .136 |
|00001f80| 20 20 20 20 20 13 0d 0a | 00 20 20 20 20 20 20 20 |     ...|.       |
|00001f90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001fa0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001fb0| 20 20 20 20 20 20 20 20 | 20 11 34 28 20 20 20 20 |        | .4(    |
|00001fc0| 20 44 32 29 0d 0b 00 11 | 31 4b 6e 6f 77 69 6e 67 | D2)....|1Knowing|
|00001fd0| 20 74 68 61 74 20 74 68 | 65 20 67 72 61 70 68 20 | that th|e graph |
|00001fe0| 70 61 73 73 65 73 20 74 | 68 72 6f 75 67 68 20 74 |passes t|hrough t|
|00001ff0| 68 65 20 70 6f 69 6e 74 | 20 11 34 21 11 31 31 2c |he point| .4!.11,|
|00002000| 20 34 11 34 53 20 11 31 | 32 11 34 21 11 31 2c 20 | 4.4S .1|2.4!.1, |
|00002010| 77 65 20 63 61 6e 20 70 | 6c 75 67 20 74 68 65 73 |we can p|lug thes|
|00002020| 65 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |e...    |        |
|00002030| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002040| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002050| 20 20 20 20 11 34 39 20 | 20 20 20 20 20 20 30 0d |    .49 |      0.|
|00002060| 0a 00 11 31 76 61 6c 75 | 65 73 20 69 6e 74 6f 20 |...1valu|es into |
|00002070| 74 68 65 20 65 71 75 61 | 74 69 6f 6e 20 74 6f 20 |the equa|tion to |
|00002080| 64 65 74 65 72 6d 69 6e | 65 20 11 33 62 20 11 31 |determin|e .3b .1|
|00002090| 61 73 20 73 68 6f 77 6e | 20 6f 6e 20 74 68 65 20 |as shown| on the |
|000020a0| 6e 65 78 74 20 70 61 67 | 65 2e 00 0c 0d 0a 00 20 |next pag|e...... |
|000020b0| 11 32 32 20 20 20 20 20 | 20 11 34 44 32 20 11 32 |.22     | .4D2 .2|
|000020c0| 32 0d 0b 00 11 31 31 20 | 20 20 20 28 34 11 34 53 |2....11 |   (4.4S|
|000020d0| 20 11 31 32 29 0d 0b 00 | 11 34 32 32 20 11 31 2b | .12)...|.422 .1+|
|000020e0| 20 11 34 32 32 32 32 32 | 32 20 20 11 31 3d 20 31 | .422222|2  .1= 1|
|000020f0| 0d 0b 00 20 11 32 32 20 | 20 20 20 20 11 31 33 36 |... .22 |    .136|
|00002100| 0d 0b 00 11 33 62 20 20 | 20 20 20 20 20 20 20 20 |....3b  |        |
|00002110| 20 20 20 20 20 11 31 13 | 0d 0a 00 20 20 20 20 20 |     .1.|...     |
|00002120| 20 20 20 20 20 31 20 20 | 20 20 20 20 20 20 33 32 |     1  |      32|
|00002130| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 11 34 32 |...     |     .42|
|00002140| 32 20 11 31 3d 20 31 20 | 2d 20 11 34 32 32 20 0d |2 .1= 1 |- .422 .|
|00002150| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 11 32 32 |..      |     .22|
|00002160| 20 20 20 20 20 20 20 11 | 31 33 36 0d 0b 00 20 20 |       .|136...  |
|00002170| 20 20 20 20 20 20 20 20 | 11 33 62 20 20 20 20 20 |        |.3b     |
|00002180| 20 20 20 20 20 20 11 31 | 13 0d 0a 00 20 20 20 20 |      .1|....    |
|00002190| 20 20 20 20 20 20 31 20 | 20 20 20 31 0d 0b 00 20 |      1 |   1... |
|000021a0| 20 20 20 20 20 20 20 20 | 20 11 34 32 32 20 11 31 |        | .422 .1|
|000021b0| 3d 20 11 34 32 20 0d 0b | 00 20 20 20 20 20 20 20 |= .42 ..|.       |
|000021c0| 20 20 20 20 11 32 32 20 | 20 20 11 31 39 0d 0b 00 |    .22 |  .19...|
|000021d0| 20 20 20 20 20 20 20 20 | 20 20 11 33 62 20 20 20 |        |  .3b   |
|000021e0| 20 20 20 11 31 13 0d 0a | 00 20 20 20 20 20 20 20 |   .1...|.       |
|000021f0| 20 20 20 20 20 11 32 32 | 0d 0b 00 20 20 20 20 20 |     .22|...     |
|00002200| 20 20 20 20 20 20 11 33 | 62 20 11 31 3d 20 39 13 |      .3|b .1= 9.|
|00002210| 20 0d 0a 00 0d 0b 00 20 | 20 20 20 20 20 20 20 20 | ...... |        |
|00002220| 20 20 11 33 62 20 11 31 | 3d 20 11 34 2b 11 31 33 |  .3b .1|= .4+.13|
|00002230| 13 0d 0a 00 0d 0b 00 54 | 68 65 72 65 66 6f 72 65 |.......T|herefore|
|00002240| 2c 20 6f 75 72 20 65 71 | 75 61 74 69 6f 6e 20 69 |, our eq|uation i|
|00002250| 73 20 0d 0a 00 20 20 20 | 20 20 20 11 32 32 20 20 |s ...   |   .22  |
|00002260| 20 20 32 0d 0b 00 20 20 | 20 20 20 11 33 78 20 20 |  2...  |   .3x  |
|00002270| 20 20 79 0d 0b 00 20 20 | 20 20 20 11 34 32 32 20 |  y...  |   .422 |
|00002280| 11 31 2b 20 11 34 32 32 | 20 11 31 3d 20 31 2e 0d |.1+ .422| .1= 1..|
|00002290| 0b 00 20 20 20 20 20 39 | 20 20 20 20 33 36 0d 0a |..     9|    36..|
|000022a0| 00 53 65 63 74 69 6f 6e | 20 34 2e 34 20 20 43 6f |.Section| 4.4  Co|
|000022b0| 6e 69 63 73 0d 0b 00 46 | 69 6e 64 20 74 68 65 20 |nics...F|ind the |
|000022c0| 63 65 6e 74 65 72 20 61 | 6e 64 20 76 65 72 74 69 |center a|nd verti|
|000022d0| 63 65 73 20 6f 66 20 74 | 68 65 20 68 79 70 65 72 |ces of t|he hyper|
|000022e0| 62 6f 6c 61 20 61 6e 64 | 20 73 6b 65 74 63 68 20 |bola and| sketch |
|000022f0| 69 74 73 20 67 72 61 70 | 68 2c 20 75 73 69 6e 67 |its grap|h, using|
|00002300| 0d 0a 00 61 73 79 6d 70 | 74 6f 74 65 73 20 61 73 |...asymp|totes as|
|00002310| 20 61 6e 20 61 69 64 2e | 0d 0a 00 20 20 20 20 20 | an aid.|...     |
|00002320| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002330| 20 11 32 32 20 20 20 20 | 32 0d 0b 00 20 20 20 20 | .22    |2...    |
|00002340| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002350| 11 31 39 11 33 78 20 20 | 11 31 2d 20 11 33 79 20 |.19.3x  |.1- .3y |
|00002360| 20 11 31 3d 20 39 0d 0a | 00 0d 0b 00 13 12 31 53 | .1= 9..|......1S|
|00002370| 4f 4c 55 54 49 4f 4e 12 | 30 0d 0a 00 0d 0b 00 57 |OLUTION.|0......W|
|00002380| 65 20 62 65 67 69 6e 20 | 62 79 20 77 72 69 74 69 |e begin |by writi|
|00002390| 6e 67 20 74 68 65 20 65 | 71 75 61 74 69 6f 6e 20 |ng the e|quation |
|000023a0| 69 6e 20 73 74 61 6e 64 | 61 72 64 20 66 6f 72 6d |in stand|ard form|
|000023b0| 2e 0d 0a 00 20 20 20 20 | 20 20 20 11 32 32 20 20 |....    |   .22  |
|000023c0| 20 20 32 0d 0b 00 20 20 | 20 20 20 11 31 39 11 33 |  2...  |   .19.3|
|000023d0| 78 20 20 11 31 2d 20 11 | 33 79 20 20 11 31 3d 20 |x  .1- .|3y  .1= |
|000023e0| 39 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |9       |        |
|000023f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 12 31 11 |        |     .1.|
|00002400| 32 47 69 76 65 6e 20 65 | 71 75 61 74 69 6f 6e 11 |2Given e|quation.|
|00002410| 31 12 30 13 0d 0a 00 20 | 20 20 20 20 20 20 11 32 |1.0.... |      .2|
|00002420| 32 20 20 20 20 32 0d 0b | 00 20 20 20 20 20 20 11 |2    2..|.      .|
|00002430| 33 78 20 20 20 20 79 0d | 0b 00 20 20 20 20 20 20 |3x    y.|..      |
|00002440| 11 34 32 32 20 11 31 2d | 20 11 34 32 32 20 11 31 |.422 .1-| .422 .1|
|00002450| 3d 20 31 0d 0b 00 20 20 | 20 20 20 20 31 20 20 20 |= 1...  |    1   |
|00002460| 20 39 20 20 20 20 20 20 | 20 20 13 0d 0a 00 20 20 | 9      |  ....  |
|00002470| 20 20 20 20 20 11 32 32 | 20 20 20 20 32 0d 0b 00 |     .22|    2...|
|00002480| 20 20 20 20 20 20 11 33 | 78 20 20 20 20 79 0d 0b |      .3|x    y..|
|00002490| 00 20 20 20 20 20 20 11 | 34 32 32 20 11 31 2d 20 |.      .|422 .1- |
|000024a0| 11 34 32 32 20 11 31 3d | 20 31 20 20 20 20 20 20 |.422 .1=| 1      |
|000024b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000024c0| 20 20 20 20 20 20 12 31 | 11 32 53 74 61 6e 64 61 |      .1|.2Standa|
|000024d0| 72 64 20 66 6f 72 6d 11 | 31 12 30 0d 0b 00 20 20 |rd form.|1.0...  |
|000024e0| 20 20 20 20 20 11 32 32 | 20 20 20 20 32 0d 0b 00 |     .22|    2...|
|000024f0| 20 20 20 20 20 20 11 31 | 31 20 20 20 20 33 20 20 |      .1|1    3  |
|00002500| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002510| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002520| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002530| 20 20 13 0d 0a 00 0d 0b | 00 57 65 20 6e 6f 74 65 |  ......|.We note|
|00002540| 20 74 68 61 74 20 74 68 | 69 73 20 69 73 20 74 68 | that th|is is th|
|00002550| 65 20 73 74 61 6e 64 61 | 72 64 20 66 6f 72 6d 20 |e standa|rd form |
|00002560| 6f 66 20 74 68 65 20 65 | 71 75 61 74 69 6f 6e 20 |of the e|quation |
|00002570| 6f 66 20 61 20 68 79 70 | 65 72 62 6f 6c 61 20 77 |of a hyp|erbola w|
|00002580| 69 74 68 0d 0a 00 63 65 | 6e 74 65 72 20 61 74 20 |ith...ce|nter at |
|00002590| 28 30 2c 20 30 29 20 77 | 68 65 72 65 20 11 33 61 |(0, 0) w|here .3a|
|000025a0| 20 11 31 3d 20 31 20 61 | 6e 64 20 11 33 62 20 11 | .1= 1 a|nd .3b .|
|000025b0| 31 3d 20 33 2e 00 0c 0d | 0b 00 20 20 20 20 20 20 |1= 3....|..      |
|000025c0| 20 20 20 20 20 20 20 11 | 32 32 0d 0b 00 11 31 42 |       .|22....1B|
|000025d0| 65 63 61 75 73 65 20 74 | 68 65 20 11 33 78 20 11 |ecause t|he .3x .|
|000025e0| 31 2d 74 65 72 6d 20 69 | 73 20 70 6f 73 69 74 69 |1-term i|s positi|
|000025f0| 76 65 2c 20 77 65 20 63 | 6f 6e 63 6c 75 64 65 20 |ve, we c|onclude |
|00002600| 74 68 61 74 20 74 68 65 | 20 74 72 61 6e 73 76 65 |that the| transve|
|00002610| 72 73 65 20 61 78 69 73 | 20 69 73 20 0d 0a 00 68 |rse axis| is ...h|
|00002620| 6f 72 69 7a 6f 6e 74 61 | 6c 20 61 6e 64 20 74 68 |orizonta|l and th|
|00002630| 65 20 76 65 72 74 69 63 | 65 73 20 6f 63 63 75 72 |e vertic|es occur|
|00002640| 20 61 74 20 28 2d 31 2c | 20 30 29 20 61 6e 64 20 | at (-1,| 0) and |
|00002650| 28 31 2c 20 30 29 2e 13 | 0d 0a 00 0d 0b 00 4b 6e |(1, 0)..|......Kn|
|00002660| 6f 77 69 6e 67 20 74 68 | 61 74 20 74 68 65 20 74 |owing th|at the t|
|00002670| 72 61 6e 73 76 65 72 73 | 65 20 61 78 69 73 20 69 |ransvers|e axis i|
|00002680| 73 20 68 6f 72 69 7a 6f | 6e 74 61 6c 20 77 65 20 |s horizo|ntal we |
|00002690| 75 73 65 20 74 68 65 20 | 76 61 6c 75 65 73 20 6f |use the |values o|
|000026a0| 66 20 11 33 61 20 11 31 | 61 6e 64 0d 0a 00 11 33 |f .3a .1|and....3|
|000026b0| 62 20 11 31 74 6f 20 64 | 65 74 65 72 6d 69 6e 65 |b .1to d|etermine|
|000026c0| 20 74 68 65 20 61 73 79 | 6d 70 74 6f 74 65 73 2e | the asy|mptotes.|
|000026d0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 11 33 62 |...     |     .3b|
|000026e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|000026f0| 31 33 0d 0b 00 20 20 20 | 20 20 11 33 79 20 11 31 |13...   |  .3y .1|
|00002700| 3d 20 11 34 2b 32 11 33 | 78 20 20 20 20 20 20 20 |= .4+2.3|x       |
|00002710| 20 20 79 20 11 31 3d 20 | 11 34 2b 32 11 33 78 0d |  y .1= |.4+2.3x.|
|00002720| 0b 00 20 20 20 20 20 20 | 20 20 20 20 61 20 20 20 |..      |    a   |
|00002730| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 31 20 |        |    .11 |
|00002740| 13 0d 0a 00 0d 0b 00 54 | 68 65 72 65 66 6f 72 65 |.......T|herefore|
|00002750| 2c 20 6f 75 72 20 73 6b | 65 74 63 68 20 63 6f 6d |, our sk|etch com|
|00002760| 65 73 20 6f 75 74 20 61 | 73 20 73 68 6f 77 6e 20 |es out a|s shown |
|00002770| 62 65 6c 6f 77 2e 0d 0a | 00 0d 0a 00 0d 0a 00 0d |below...|........|
|00002780| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 14 6b 2d |..      |     .k-|
|00002790| 35 2d 33 2d 36 2e 68 69 | 14 32 30 14 31 38 14 31 |5-3-6.hi|.20.18.1|
|000027a0| 38 14 38 14 0d 0a 00 0d | 0a 00 53 65 63 74 69 6f |8.8.....|..Sectio|
|000027b0| 6e 20 34 2e 34 20 20 43 | 6f 6e 69 63 73 0d 0b 00 |n 4.4  C|onics...|
|000027c0| 46 69 6e 64 20 61 6e 20 | 65 71 75 61 74 69 6f 6e |Find an |equation|
|000027d0| 20 6f 66 20 74 68 65 20 | 68 79 70 65 72 62 6f 6c | of the |hyperbol|
|000027e0| 61 20 77 69 74 68 20 63 | 65 6e 74 65 72 20 61 74 |a with c|enter at|
|000027f0| 20 74 68 65 20 6f 72 69 | 67 69 6e 2c 20 76 65 72 | the ori|gin, ver|
|00002800| 74 69 63 65 73 20 61 74 | 0d 0a 00 28 11 34 2b 11 |tices at|...(.4+.|
|00002810| 31 33 2c 20 30 29 20 61 | 6e 64 20 61 73 79 6d 70 |13, 0) a|nd asymp|
|00002820| 74 6f 74 65 73 20 11 33 | 79 20 11 31 3d 20 11 34 |totes .3|y .1= .4|
|00002830| 2b 11 31 32 11 33 78 11 | 31 2e 0d 0a 00 0d 0b 00 |+.12.3x.|1.......|
|00002840| 13 12 31 53 4f 4c 55 54 | 49 4f 4e 12 30 0d 0a 00 |..1SOLUT|ION.0...|
|00002850| 0d 0b 00 53 69 6e 63 65 | 20 74 68 65 20 76 65 72 |...Since| the ver|
|00002860| 74 69 63 65 73 20 6f 63 | 63 75 72 20 61 74 20 28 |tices oc|cur at (|
|00002870| 2d 33 2c 20 30 29 20 61 | 6e 64 20 28 33 2c 20 30 |-3, 0) a|nd (3, 0|
|00002880| 29 20 77 65 20 6b 6e 6f | 77 20 74 68 61 74 20 74 |) we kno|w that t|
|00002890| 68 65 20 74 72 61 6e 73 | 76 65 72 73 65 0d 0a 00 |he trans|verse...|
|000028a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000028b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000028c0| 20 20 20 20 20 20 20 20 | 20 11 32 32 20 20 20 20 |        | .22    |
|000028d0| 32 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |2...    |        |
|000028e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000028f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 33 78 20 |        |    .3x |
|00002900| 20 20 20 79 0d 0b 00 11 | 31 61 78 69 73 20 69 73 |   y....|1axis is|
|00002910| 20 68 6f 72 69 7a 6f 6e | 74 61 6c 20 61 6e 64 20 | horizon|tal and |
|00002920| 77 65 20 68 61 76 65 20 | 74 68 65 20 66 6f 72 6d |we have |the form|
|00002930| 20 11 34 32 32 20 11 31 | 2d 20 11 34 32 32 20 11 | .422 .1|- .422 .|
|00002940| 31 3d 20 31 20 77 68 65 | 72 65 20 11 33 61 20 11 |1= 1 whe|re .3a .|
|00002950| 31 3d 20 33 2e 0d 0b 00 | 20 20 20 20 20 20 20 20 |1= 3....|        |
|00002960| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002970| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002980| 20 11 32 32 20 20 20 20 | 32 0d 0b 00 20 20 20 20 | .22    |2...    |
|00002990| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000029a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000029b0| 20 20 20 20 11 33 61 20 | 20 20 20 62 20 20 20 20 |    .3a |   b    |
|000029c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 31 |        |      .1|
|000029d0| 13 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|000029e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000029f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002a00| 20 20 20 20 20 20 20 20 | 20 20 20 11 33 62 0d 0b |        |   .3b..|
|00002a10| 00 11 31 57 65 20 61 6c | 73 6f 20 6b 6e 6f 77 20 |..1We al|so know |
|00002a20| 74 68 61 74 20 74 68 65 | 20 61 73 79 6d 70 74 6f |that the| asympto|
|00002a30| 74 65 73 20 61 72 65 20 | 6f 66 20 74 68 65 20 66 |tes are |of the f|
|00002a40| 6f 72 6d 20 11 33 79 20 | 11 31 3d 20 11 34 2b 20 |orm .3y |.1= .4+ |
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+--------+-------------------------+-------------------------+--------+--------+