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Find curve with tangent and normal lines that create a triangle with given area

  1. Oct 9, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the implicit equation of the curve that goes through the point (3, 1) and whose tangent and normal lines always form with the x axis a triangle whose area is equal to the slope of the tangent line. Assume y` > 0 and y > 0.


    2. Relevant equations

    Hint: ∫( √(a^2 - u^2) / u du = √(a^2-u^2) - a*ln | [a+√(a^2-u^2)] / u | + C
    (sorry, I don't know how to use the math writer yet)

    3. The attempt at a solution

    This is a question from an introductory differential equations class. I have absolutely no idea how to do this! I haven't really gotten anywhere yet. This is what I've done:

    let f(x) denote the curve we're looking for. Then the tangent line will have equation:
    y_t = df/dx * x + C
    Normal line will have equation y_n = -1/(df/dx) * x + k

    Together they will form a triangle with area = df/dx, at any point on f(x). I wanted to find an expression for area in terms of df/dx, simplify it, and solve the resulting differential equation, but I can't figure out a DE for the area! I'm getting very frustrated, as we've never been shown a question like this in lecture, and I can't find any examples in my textbook.

    Help would be very much appreciated!
     
  2. jcsd
  3. Oct 10, 2011 #2
    Moore's 201 class at uvic? same boat...

    the only useful thing ive written down is dT/dx = (1/2)(T)(N)

    its dT/dx because its equal to the slope of the tangent line

    Please post back with any progress you make and ill do the same
     
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