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Mimi_x3 Group Title

If the chord of contact of the tangents to a parabola x^2=4ay, from an external point T(x1,y1) meets the directrix at R, prove that RT subtends a right angle at the focus.

  • 3 years ago
  • 3 years ago

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  1. perl Group Title
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    mmmmm, you werent kidding around

    • 3 years ago
  2. Mimi_x3 Group Title
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    Is this easy for you ? xD

    • 3 years ago
  3. perl Group Title
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    i will have to think about it

    • 3 years ago
  4. perl Group Title
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    the directrix is that line under the parabola

    • 3 years ago
  5. Mimi_x3 Group Title
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    lols, you were thinking that its easy ? xD

    • 3 years ago
  6. Mimi_x3 Group Title
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    Yup, y=-a

    • 3 years ago
  7. perl Group Title
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    what class is this ?

    • 3 years ago
  8. Mimi_x3 Group Title
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    I live in Australia, so its kinda different, there's only a maths class xD

    • 3 years ago
  9. perl Group Title
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    sorry wrong question, have you done calculus

    • 3 years ago
  10. Mimi_x3 Group Title
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    Yes I have, how did you know ? xD

    • 3 years ago
  11. Mimi_x3 Group Title
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    What grade are you in ?

    • 3 years ago
  12. perl Group Title
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    it will definitely help

    • 3 years ago
  13. perl Group Title
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    im a sophomore in college

    • 3 years ago
  14. Mimi_x3 Group Title
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    Cool~ college, and you havent learnt that ?

    • 3 years ago
  15. perl Group Title
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    not specifically, but i bet i can do this

    • 3 years ago
  16. Mimi_x3 Group Title
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    lols, then try hehe

    • 3 years ago
  17. perl Group Title
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    im trying to make a diagram, im not even sure i understand the question yet

    • 3 years ago
  18. perl Group Title
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    ok you have an external pt, and you want the tangent through it

    • 3 years ago
  19. Mimi_x3 Group Title
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    Yeah, I don't really know where's the external point is, thats why im lost =/

    • 3 years ago
  20. perl Group Title
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    ok first thing is first. the chord of contact means something special

    • 3 years ago
  21. Mimi_x3 Group Title
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    Something special ? What do you mean ?

    • 3 years ago
  22. perl Group Title
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    there is a picture of a chord of contacts http://www.nointrigue.com/docs/notes/maths/maths_parametrics.pdf

    • 3 years ago
  23. perl Group Title
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    well given an external point there are two tangents from that point to the parabola

    • 3 years ago
  24. perl Group Title
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    now where the tangents intersect the parabola, now draw a chord through those two points

    • 3 years ago
  25. perl Group Title
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    where the tangents meet the parabola, the two tangents

    • 3 years ago
  26. Mimi_x3 Group Title
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    wow~ nice pdf file, its detailed on parametrics ty

    • 3 years ago
  27. Mimi_x3 Group Title
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    But where is the external point ?

    • 3 years ago
  28. perl Group Title
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    anywhere

    • 3 years ago
  29. Mimi_x3 Group Title
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    Can't it have to be specific

    • 3 years ago
  30. perl Group Title
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    in the pdf, the point is below the parabola

    • 3 years ago
  31. Mimi_x3 Group Title
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    It can be but it doesn't make sense when it says RT subtends a right angle at the focus.

    • 3 years ago
  32. perl Group Title
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    like this

    • 3 years ago
  33. Mimi_x3 Group Title
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    like this ?

    • 3 years ago
  34. perl Group Title
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    |dw:1319452686670:dw|

    • 3 years ago
  35. Mimi_x3 Group Title
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    I know but where's the external point ?

    • 3 years ago
  36. perl Group Title
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    |dw:1319452750463:dw|

    • 3 years ago
  37. Mimi_x3 Group Title
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    Imaginary ?

    • 3 years ago
  38. perl Group Title
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    |dw:1319452790390:dw|

    • 3 years ago
  39. Mimi_x3 Group Title
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    Is the parabola imaginery or somethinbg ?

    • 3 years ago
  40. Mimi_x3 Group Title
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    imaginary* something*

    • 3 years ago
  41. perl Group Title
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    |dw:1319452894222:dw|

    • 3 years ago
  42. Mimi_x3 Group Title
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    I don't get your diagrams ? sorry

    • 3 years ago
  43. perl Group Title
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    |dw:1319453015665:dw|

    • 3 years ago
  44. Mimi_x3 Group Title
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    I don't get it ? >_<

    • 3 years ago
  45. perl Group Title
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    yeah the diagram is a beast

    • 3 years ago
  46. Mimi_x3 Group Title
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    lols, why is there like 3 digrams ? i dont get it >_<

    • 3 years ago
  47. perl Group Title
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    |dw:1319453240562:dw|

    • 3 years ago
  48. Mimi_x3 Group Title
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    lols, this is not 3D trig xD

    • 3 years ago
  49. perl Group Title
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    can you do better ?

    • 3 years ago
  50. Mimi_x3 Group Title
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    Nope, i dont know how to start, the problem is where is T

    • 3 years ago
  51. perl Group Title
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    just a point below the parabola

    • 3 years ago
  52. perl Group Title
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    like in that pdf

    • 3 years ago
  53. Mimi_x3 Group Title
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    But it won't make sense, and i dont know how to prove

    • 3 years ago
  54. perl Group Title
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    have faith

    • 3 years ago
  55. Mimi_x3 Group Title
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    have faith ?

    • 3 years ago
  56. perl Group Title
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    i think they mean the segment R T

    • 3 years ago
  57. perl Group Title
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    yes have faith when you draw it that it will all come into place, the problem will unravel

    • 3 years ago
  58. perl Group Title
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    i am atheist, what did you think i mean ?

    • 3 years ago
  59. Mimi_x3 Group Title
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    idk, never mind then i will work it out myself

    • 3 years ago
  60. perl Group Title
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    |dw:1319453605908:dw|

    • 3 years ago
  61. Mimi_x3 Group Title
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    ty (:

    • 3 years ago
  62. perl Group Title
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    im working on it

    • 3 years ago
  63. perl Group Title
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    would be nice if they gave you a diagram

    • 3 years ago
  64. Mimi_x3 Group Title
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    Yeah i know, none of the hard questions have diagrams =/

    • 3 years ago
  65. perl Group Title
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    i think RT is the line segment though

    • 3 years ago
  66. Mimi_x3 Group Title
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    hm idk

    • 3 years ago
  67. perl Group Title
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    |dw:1319454238070:dw|

    • 3 years ago
  68. agdgdgdgwngo Group Title
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    \[\text{An equation of the parabola with focus }(0,p)\text{ and directrix }y = -p\text{ is}\] \[x^2=4py\]

    • 3 years ago
  69. agdgdgdgwngo Group Title
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    so the directrix would be y = -a in your question :-D

    • 3 years ago
  70. perl Group Title
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    yes

    • 3 years ago
  71. Mimi_x3 Group Title
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    Yeah, but im having problem proving

    • 3 years ago
  72. agdgdgdgwngo Group Title
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    If the chord of contact of the tangents to a parabola x² = 4ay, from an external point T(x1, y1) meets the directrix at R, prove that RT subtends a right angle at the focus. chord of contact of tangents to parabola is xx1=2a(y+y1) directrix is y=-a substitute into that, x=[2a(-a+y1)]/x1 y=-a R([2a(-a+y1)]/x1,-a) let S be the focus d/dx RS = 2a/ -[2a(-a+y1)]/x1= x1/(a-y1) d/dx TS=(y1-a)/x1 d/dx RS * d/dx TS= -1

    • 3 years ago
  73. Mimi_x3 Group Title
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    lols, you got it from boredofstudies right ?

    • 3 years ago
  74. agdgdgdgwngo Group Title
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    yes... :-P

    • 3 years ago
  75. perl Group Title
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    how did you get chord of tangents equation

    • 3 years ago
  76. perl Group Title
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    hey, can you draw a diagram then of whats going on, please post the link

    • 3 years ago
  77. Mimi_x3 Group Title
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    Yeah but the problem is that I don't understand the solution

    • 3 years ago
  78. agdgdgdgwngo Group Title
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    it helps to draw a diagram.

    • 3 years ago
  79. perl Group Title
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    can you post the link to the solution please , so i can read it

    • 3 years ago
  80. agdgdgdgwngo Group Title
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    http://community.boredofstudies.org/showthread.php?t=192825&page=1

    • 3 years ago
  81. Mimi_x3 Group Title
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    But what's xx1 ? im trying to figure it out

    • 3 years ago
  82. agdgdgdgwngo Group Title
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    http://wiki.answers.com/Q/Find_the_equation_of_chord_of_contact_tangent_to_the_parabola

    • 3 years ago
  83. Mimi_x3 Group Title
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    oh i get it, but where did y=1/2(p+q)x-apq come from ?

    • 3 years ago
  84. agdgdgdgwngo Group Title
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    xx1 must mean something like \[x_1\times x= 2a(y+y_1)\]

    • 3 years ago
  85. Mimi_x3 Group Title
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    where did 2a come from ?

    • 3 years ago
  86. agdgdgdgwngo Group Title
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    good question

    • 3 years ago
  87. agdgdgdgwngo Group Title
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    \[parabola: x^2 = 4py\]\[a = \frac{1}{4p}\]

    • 3 years ago
  88. Mimi_x3 Group Title
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    no, isn't it x^2 = 4ay y = x^2/4a y' = 2x/4a = x/2a

    • 3 years ago
  89. perl Group Title
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    |dw:1319455562515:dw|

    • 3 years ago
  90. perl Group Title
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    i wish there were other colors, but from T draw two tangents to the parabola

    • 3 years ago
  91. agdgdgdgwngo Group Title
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    perl's diagram works.... but how do we get there using calculus and derivatives if we don't even know the equation for chord of contact of the tangents to a parabola

    • 3 years ago
  92. Mimi_x3 Group Title
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    oh the equation of the tangent to the parabola is y=px-ap^2 right ?

    • 3 years ago
  93. perl Group Title
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    |dw:1319455702848:dw|

    • 3 years ago
  94. perl Group Title
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    they are given here http://www.nointrigue.com/docs/notes/maths/maths_parametrics.pdf

    • 3 years ago
  95. perl Group Title
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    the chords of tangents formula

    • 3 years ago
  96. agdgdgdgwngo Group Title
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    now it all makes sense

    • 3 years ago
  97. perl Group Title
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    |dw:1319455885258:dw|

    • 3 years ago
  98. perl Group Title
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    i dont get that parametrisation, is it because they want x^2 = 4ay ? , what is p

    • 3 years ago
  99. Mimi_x3 Group Title
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    Doesn't it involve parameters as well P(2ap,ap^2)

    • 3 years ago
  100. agdgdgdgwngo Group Title
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    they want something like the quotient of the slopes between 2 lines is equal to -1

    • 3 years ago
  101. agdgdgdgwngo Group Title
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    and one of those lines should be RT

    • 3 years ago
  102. perl Group Title
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    |dw:1319456032235:dw|

    • 3 years ago
  103. perl Group Title
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    crap, it missed the R it is the right bottom corner

    • 3 years ago
  104. Mimi_x3 Group Title
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    Im confused, is parameters involved ?

    • 3 years ago
  105. agdgdgdgwngo Group Title
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    no parameter seems to be involved :-(

    • 3 years ago
  106. perl Group Title
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    actually a is the parameter, you have to be careful, this word parameter is used in different senses

    • 3 years ago
  107. perl Group Title
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    , for instance in y = mx + b , m and b are parameters . now another sense of parameter is used when we 'parametrize' an equation. this means we introduce a third variable, often t

    • 3 years ago
  108. Mimi_x3 Group Title
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    isn't parameters P(2ap,ap^2) So there must be a point for the parameters ?

    • 3 years ago
  109. agdgdgdgwngo Group Title
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    meh, I assumed m and b to be constant

    • 3 years ago
  110. perl Group Title
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    right, but they are considered 'parameters'

    • 3 years ago
  111. Mimi_x3 Group Title
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    but its not a straight line so it cant be y=mx + b

    • 3 years ago
  112. perl Group Title
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    in the sense that they change your line . so for x^2 = 4ay, the parameter is a since that produces different parabolas by tweaking a

    • 3 years ago
  113. perl Group Title
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    think of in this sense a parameter is your tweaking constant

    • 3 years ago
  114. agdgdgdgwngo Group Title
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    :-(

    • 3 years ago
  115. perl Group Title
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    the parameter is 'a'*

    • 3 years ago
  116. agdgdgdgwngo Group Title
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    I thought a was just a slope

    • 3 years ago
  117. Mimi_x3 Group Title
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    Im lost =/ Parameter P(2ap,ap^2)_

    • 3 years ago
  118. perl Group Title
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    changing 'a' changes your parabola. but for the line y = mx + b, changing m,b changes your line

    • 3 years ago
  119. perl Group Title
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    these are your tweaking values .

    • 3 years ago
  120. perl Group Title
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    I dont like their use of p, scratch that and use t instead

    • 3 years ago
  121. perl Group Title
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    ok they are using 'parametric equations' , x = f(t), y = g(t)

    • 3 years ago
  122. perl Group Title
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    so x = 2at, y = at^2

    • 3 years ago
  123. agdgdgdgwngo Group Title
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    I like to use matrices and vectors.

    • 3 years ago
  124. Mimi_x3 Group Title
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    Yeah thats in cartesian form

    • 3 years ago
  125. perl Group Title
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    ok, use a matrix or vector, id like to see that

    • 3 years ago
  126. perl Group Title
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    mimi, when you remove t, you get cartesian equation

    • 3 years ago
  127. Mimi_x3 Group Title
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    matrix or vector ?

    • 3 years ago
  128. Mimi_x3 Group Title
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    Oh yeah my bad i forgot xD

    • 3 years ago
  129. agdgdgdgwngo Group Title
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    http://img392.imageshack.us/img392/8476/lach11n72fb.jpg

    • 3 years ago
  130. Mimi_x3 Group Title
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    I haven't learnt that complicated stuff yet =/

    • 3 years ago
  131. agdgdgdgwngo Group Title
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    you will learn it soon.

    • 3 years ago
  132. Mimi_x3 Group Title
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    Hope not looks hard already =/

    • 3 years ago
  133. perl Group Title
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    thats curve fitting, no big deal

    • 3 years ago
  134. agdgdgdgwngo Group Title
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    you will find it easier than conics

    • 3 years ago
  135. Mimi_x3 Group Title
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    Whats conics ?

    • 3 years ago
  136. perl Group Title
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    its a quick way to fit a quadratic sequence, cubic, etc

    • 3 years ago
  137. perl Group Title
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    mimi, what we are doing is conics, oh dea

    • 3 years ago
  138. agdgdgdgwngo Group Title
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    conic sections: parabolas, circles, ellipses, hyperbolas, stuff.

    • 3 years ago
  139. perl Group Title
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    oh dear *

    • 3 years ago
  140. agdgdgdgwngo Group Title
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    you do them along parametric equations usually.

    • 3 years ago
  141. perl Group Title
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    agdg, put the proper definition on board ;)

    • 3 years ago
  142. agdgdgdgwngo Group Title
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    well, they're what you get when you cut a cone.

    • 3 years ago
  143. Mimi_x3 Group Title
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    LOL, this is not conics

    • 3 years ago
  144. perl Group Title
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    a conic has a specific algebraic equation in x and y , quadratic Ax^2 + Bxy + cY^2 + dy + ex + f = 0

    • 3 years ago
  145. perl Group Title
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    yes it is, a parabola is a conic

    • 3 years ago
  146. perl Group Title
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    a conic is a family of curves, as agdg pointed out

    • 3 years ago
  147. Mimi_x3 Group Title
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    But Ax^2 + Bxy + cY^2 + dy + ex + f = 0 ?

    • 3 years ago
  148. agdgdgdgwngo Group Title
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    forget the algebra and xs and ys and f(x)s, you will understand them really well once you picture them geometrically

    • 3 years ago
  149. agdgdgdgwngo Group Title
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    http://mathworld.wolfram.com/ConicSection.html

    • 3 years ago
  150. perl Group Title
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    right

    • 3 years ago
  151. Mimi_x3 Group Title
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    Um, this doesnt look like a conic

    • 3 years ago
  152. perl Group Title
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    agdg, whats the nth term of this quadratic sequence 3,5,10,18...

    • 3 years ago
  153. perl Group Title
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    lets test your matrix skills

    • 3 years ago
  154. agdgdgdgwngo Group Title
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    hmmm... is it infinity?

    • 3 years ago
  155. perl Group Title
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    no, the nth term

    • 3 years ago
  156. perl Group Title
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    give me a formula for the sequence

    • 3 years ago
  157. perl Group Title
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    obviously it diverges, ;)

    • 3 years ago
  158. agdgdgdgwngo Group Title
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    its \[2^n + 2^\frac{n}{2}\]

    • 3 years ago
  159. perl Group Title
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    no, lol

    • 3 years ago
  160. agdgdgdgwngo Group Title
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    wait! it's \[\frac{2^n+2^{\frac{n}{2}}}{2}\]

    • 3 years ago
  161. perl Group Title
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    its a quadratic sequence, like in your matrix

    • 3 years ago
  162. agdgdgdgwngo Group Title
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    how do we find it :-D

    • 3 years ago
  163. agdgdgdgwngo Group Title
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    so what's the 5th term

    • 3 years ago
  164. perl Group Title
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    ok so first we find the differences

    • 3 years ago
  165. perl Group Title
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    the first row of differences is 2 , 5, 8, 11 , ... the second row of differences is 3, 3, 3, ... so you know its a quadratic sequence

    • 3 years ago
  166. agdgdgdgwngo Group Title
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    that looks like a long matrix

    • 3 years ago
  167. perl Group Title
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    3, 5, 10, 18, 29 ... thats the sequence

    • 3 years ago
  168. perl Group Title
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    it should fit f(n) = an^2 + bn + c

    • 3 years ago
  169. agdgdgdgwngo Group Title
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    I guess that works.

    • 3 years ago
  170. agdgdgdgwngo Group Title
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    how do we multiply matrices?

    • 3 years ago
  171. perl Group Title
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    are you asking in general ?

    • 3 years ago
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