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Another Turning point question

  1. Feb 2, 2013 #1
    Find the equation of the tangent to the curve xy=4 at the point P whose coordinates are (2t,2/t). If O is the origin and the tangent at P meets the x-axis at A and the y axis at B, prove
    (a) that P is the midpoint of AB
    (b) that the area of Triangle OAB is the same for all positions of P.
    Find the the equations of the normals to the curve xy=4 which are parallel to the line
    4x-y-2=0.


    I got the equation of the of the tangent at P to be yt2 - 4t -x = 0

    Not sure how to do a and b I don't really have much information about AB other than it's points like on P basically.
     
  2. jcsd
  3. Feb 2, 2013 #2

    mfb

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    The y-coordinate of A is 0. It lies on the tangent, so it satisfies yt2 - 4t -x = 0. What is its x-coordinate?
     
  4. Feb 2, 2013 #3
    x is -4t
     
  5. Feb 2, 2013 #4

    SammyS

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    Now that you have the coordinates for point A, find the coordinates for point B. (The x coordinate for point B is 0.)
     
  6. Feb 2, 2013 #5
    B is (0, 4/t) oh and when you work it out the mid point is in fact P... to do part b now do I find the length of the lines and work out the area?
     
  7. Feb 2, 2013 #6

    SammyS

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    Draw a sketch .

    The result should be pretty obvious.
     
  8. Feb 2, 2013 #7
    34fdwsm.png

    So no calculations are needed to prove it?

    How would I do the final part if I don't know any points for the normals?
     
  9. Feb 2, 2013 #8

    SammyS

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    I just noticed that must be an error in your equation for the tangent line. Assuming that t is positive, both intercepts should be positive.

    You have the wrong sign on x.
     
  10. Feb 2, 2013 #9
    This should be okay now right?

    dxicxw.png
     
  11. Feb 2, 2013 #10

    rollingstein

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    Area = 1/2 * 4t * 4/t = 8

    Hence constant.
     
  12. Feb 2, 2013 #11

    rollingstein

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    yt2 - 4t + x = 0
     
  13. Feb 2, 2013 #12
    Yeah now I see it, thanks. But the final part of the question
    Find the the equations of the normals to the curve xy=4 which are parallel to the line
    4x-y-2=0.

    How would I do this without having any points for the normals?
     
  14. Feb 2, 2013 #13

    rollingstein

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    I get the normal as y=4x-15
     
  15. Feb 2, 2013 #14
    How did you calculate it I don't understand we don't have any points.

    :S
     
  16. Feb 2, 2013 #15

    rollingstein

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    I'm not sure I am right.

    But, Slope * Slope_normal = -1

    You know Slope of curve so find slope of normal.

    Then equate it to slope of given line to find where the normal lies.
     
  17. Feb 2, 2013 #16
    slope of curve = -4x^2 = equation of given line 4x-2

    solve for x? then get y from equation of the curve then find equation of normal?

    IGNORE this makes no sense.
     
    Last edited: Feb 2, 2013
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