Question: Sliding ellipse problem

Has anyone solved this problem from an older Putnam paper?

An ellipse sitting in the first quadrant with its major axis parallel to the x axis is tangent to the positive x and y axes.

It slides clockwise within the first quadrant while maintaining tangency to both positive axes until its major axis is parallel to the y axis.

Prove that the locus of its centre is the arc of a circle.

I have crudely animated this motion by sliding the axes around the stationary ellipse. Is there a more elegant animation which slides the ellipse against stationary axes?

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