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Horizontal and Vertical Tangents for Polar Equation

  1. Jan 16, 2014 #1
    1. The problem statement, all variables and given/known data
    Find the points at which the following polar equation has horizontal and vertical tangents:

    r2 = 4cos(2θ)


    2. Relevant equations

    [itex]\frac{dy}{dx}[/itex] = [itex]\frac{r'(θ)sinθ + r(θ)cosθ}{r'(θ)cosθ - r(θ)sinθ}[/itex]

    Horizontal Tangent: [itex]\frac{dy}{dθ}[/itex] = 0; [itex]\frac{dx}{dθ}[/itex] ≠ 0
    Vertical Tangent: [itex]\frac{dx}{dθ}[/itex] = 0; [itex]\frac{dy}{dθ}[/itex] ≠ 0

    3. The attempt at a solution
    There is no "clean" way of solving for r (because of the +/- sqrt) so that I could find r'(θ) to use in the formula. So, I figured I would use implicit differentiation:

    r2 = 4cos(2θ)

    2r[itex]\frac{dr}{dθ}[/itex] = -8sin(2θ)

    [itex]\frac{dr}{dθ}[/itex] = [itex]\frac{-4sin(2θ)}{r}[/itex]

    But, when I plug it in the formula, I now have an r and a θ when I set the numerator and denominator of [itex]\frac{dy}{dx}[/itex] equal to zero (separately of course). I don't know what to do when I have both variables like that and I'm trying to solve for θ. How should I be approaching this problem? Have I done this problem right so far? If so, what do I do next? If not, any suggestions? Please be clear. I can't find this problem answered clearly anywhere on the internet.
    Thank you
     
  2. jcsd
  3. Jan 16, 2014 #2
    For some reason the text isn't showing up right on my computer after posting. Let me know if its the same for you so I can edit and try to fix it.
     
  4. Jan 16, 2014 #3

    haruspex

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    You should have an equation involving r and θ (no derivatives around). Your original equation for the curve also has those two variables. Two equations, two unknowns.
    If still stuck, please post all your working.
     
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