Equation of the locus of centre of the ellipse?

Question

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

Answer 1

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.