Problem #57

Two ellipses have semimajor axes of length 2 and semiminor axes of length 1. In one ellipse, the semimajor axis is horizontal; in the other, the semiminor axis is horizontal. The two ellipses are tangent and the horizontal axes (semimajor for one, semiminor for the other) lie on the same line. The ellipses lie on opposite sides of their common tangent line [see the figure below].
What is the radius of the smallest circle that contains both ellipses?

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