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Homework Help: Find the curve given the tangent

  1. Jun 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Given that the tangent to the curve [itex] c(t) [/itex] at any point on the curve is [itex] T(t) = (-sin(t), cos(t) )[/itex], find [itex] c(t) [/itex] if the curve passes through the point [itex] (0,0) [/itex].


    3. The attempt at a solution

    I try to let
    [itex] c(t) = ( x(t), y(t) ) [/itex]
    Then
    [itex] c'(t) = ( x'(t), y'(t) ) [/itex]
    [itex]| c'(t) | = \sqrt{[x'(t)]^2 + [y'(t)]^2 } [/itex]
    And
    [itex] T(t) = \frac{c'(t)}{|c'(t)|} [/itex]

    However this is complicated and consequently I am not sure how to solve it. I am also not sure how to "use" the point given since (0, 0) correspond the the values x and y respectively rather than t.

    Thanks.
     
  2. jcsd
  3. Jun 5, 2010 #2

    rock.freak667

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    Homework Helper

    If you had c'(t) =<2t,1> then c(t) = <t2+A,t+B> where A,B=constant.


    But you have

    c'(t)/|c'(t)| = <-sint,cost>

    so what is c(t) ? (note that |c'(t)| is just a constant)

    At (0,0), t=0. So what is c(t) now?
     
  4. Jun 5, 2010 #3
    c(t) = (cos(t) + C, sin(t) + K)
    t= 0, point is (0,0)

    So x(t) = 1 + C = 0, C = -1
    And y(t) = 0 + K, K=0

    So c(t) = (cos(t) -1 , sin(t) ) ?
     
  5. Jun 5, 2010 #4

    rock.freak667

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    Homework Helper

    Yes but you forgot out |c'(t)| which is the distance from the center to any tangent. In this case it would just be the same as |T(t)|
     
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