# Thread: parametrized curve - ellipse

1. ## 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 ?!

2. ## Re: parametrized curve - ellipse

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