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Math Help - parametrized curve - ellipse

  1. #1
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    parametrized curve - ellipse

    Let γ : R → R2 be the plane curve given by γ(t) = (3 tanh t, 2 sech t). Show that the
    image of γ is contained in an ellipse, and determine the equation of the ellipse. Is the
    image the whole ellipse?

    I can't work out where to start, the only thing I can think is that I have to use tanh^2 x + sech^2 x =1, but I'm not sure how ?!
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    Re: parametrized curve - ellipse

    \displaystyle \begin{align*} x = 3\tanh{(t)} \implies \frac{x}{3} = \tanh{(t)} \implies \frac{x^2}{9} = \tanh^2{(t)} \end{align*} and \displaystyle \begin{align*} y = 2\,\textrm{sech}\,{(t)} \implies \frac{y}{2} = \textrm{sech}\,{(t)} \implies \frac{y^2}{4} = \textrm{sech}^2\,{(t)} \end{align*}. Since you know that \displaystyle \begin{align*} \tanh^2{(t)} + \textrm{sech}^2\,{(t)} \equiv 1 \end{align*}, that must mean \displaystyle \begin{align*} \frac{x^2}{9} + \frac{y^2}{4} = 1 \end{align*} is the equation of your ellipse.
    Thanks from algebra123
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