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Math Help - Cubics

  1. #1
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    Cubics

    (2.1) Examples of parametrised cubics. Some plane cubic curves can be parametrised, just as the conics:

    Nodal cubic C: (y^2=x^3+x^2) \subset \mathbb{R}^2 is the image of the map \varphi: \mathbb{R}^1 \to \mathbb{R}^2 given by t\mapsto (t^2-1, t^3-t)

    Cuspidal cubic. C: (y^2 = x^3)\subset \mathbb{R}^2 is the image of \varphi:\mathbb{R}^1 \to \mathbb{R}^2 given by t\mapsto (t^2,t^3)


    (1) Let C: (y^2 = x^3 + x^2) \subset \mathbb{R}^2. Show that a varible line though (0,0) meets C at one further point, and hence deduce the parametrisation of C given in (2.1). Do the same for (y^2 = x^3) and (x^3 = y^3-y^4).

    If anyone could show me how to do any of those in (1), I would really appreciate it.. I have no idea how to tackle this problem. Thanks!
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  2. #2
    MHF Contributor
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    The first step is to determine what a line through the origin looks like.
    The equation of this line will be y = kx for some k. Then:

    y^2 = x^3 + x^2

    Substituting for y:

    (kx)^2 = x^3 + x^2

    x^3 = (kx)^2 - x^2

    x^3 = k^2x^2 - x^2

    x^3 = (k^2 - 1)x^2

    For x \neq 0:

    x = k^2 - 1

    And since y = kx:

    y = k(k^2 - 1)
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