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An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?

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By this, do you mean that the ellipse is rotated and translated continuously in the plane, so that at all times each of the two perpendicular lines is tangent to the ellipse, until the ellipse traverses through all possible such positions? –  Cameron Buie Aug 1 '12 at 19:56

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Given the parametric equation of a rotated ellipse $$ x(t)=x_0+a\cos\theta\cos{t}-b\sin\theta\sin{t}\\ y(t)=y_0+b\cos\theta\sin{t}+a\sin\theta\cos{t} $$ the conditions $\dot{x}(t)=x(t)=0$ for the contact point to the vertical line give $$ x_0=\sqrt{a^2\cos^2\theta+b^2\sin^2\theta} $$ and from $\dot{y}(t)=y(t)=0$ $$ y_0=\sqrt{a^2\sin^2\theta+b^2\cos^2\theta} $$ Here is an animated graphics.

enter image description here

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+1, very nice! I didn't think the solution would be that simple. It's worth mentioning, though, that these points lie on the circle $x_0^2+y_0^2=a^2+b^2$ centred at the origin. –  joriki Aug 1 '12 at 21:13
    
@joriki: Really interesting observation that I missed. –  enzotib Aug 1 '12 at 21:32
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In short, the circle is a point-glissette of the ellipse. See also this animation from here. –  J. M. Aug 2 '12 at 4:46
    
@J.M.... that web page... it's hypnotizing... *swoons* –  Rahul Aug 2 '12 at 8:30
    
@Rahul, shoulda put in a warning, I guess... –  J. M. Aug 2 '12 at 11:53

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