An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?
Equation of the locus of centre of the ellipse?
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$\begingroup$
conic-sections
plane-curves
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0By 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? – 2012-08-01
1 Answers
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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.
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0i think that the solution is wrong because the locus stated above is not of the path traced by center of ellipse whereas it is the locus of the director circle (a circle formed by the points where two perpendicular tangent lines to the curve intersect) therefore it is not correct according to me – 2016-01-31