May 10th, 2017, 01:41 AM  #1 
Member Joined: Jan 2016 From: Blackpool Posts: 95 Thanks: 2  ellipse question
Let γ : R → R 2 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? 
May 10th, 2017, 03:33 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1618 
As tanh²t + sech²t ≡ 1, an equation for the ellipse is (x/3)^2 + (y/2)^2 = 1. As sech t > 0, the image isn't the whole ellipse. 

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