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Math Help - Find the condition for line to touch a curve?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find the condition for line to touch a curve?

    Find the condition that the line x\cos \alpha + y\sin \alpha = p may touch the curve

    \left(\frac{x - a}{a}\right)^{\frac{n}{n - 1}} + \left(\frac{y - b}{b}\right)^{\frac{n}{n - 1}} = 1
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by fardeen_gen View Post
    Find the condition that the line x\cos \alpha + y\sin \alpha = p may touch the curve

    \left(\frac{x - a}{a}\right)^{\frac{n}{n - 1}} + \left(\frac{y - b}{b}\right)^{\frac{n}{n - 1}} = 1
    Assume n>1,\ a, b>0

    The second curve is a hyper-ellipse, and the condition of tangency is that the equation you get by substituting y from the equation for the line into the equation of the hyper-ellipse has a single real root.

    CB
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