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 ?!

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.