You are given a fixed ellipse with equation
*x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1
and a varying circle with equation *x*^{2} + *y*^{2} = *r*^{2}.
Let *A* be a point on the circle and *B* be a point on the ellipse such that
the line through *A* and *B* is a common tangent to the circle and to the
ellipse (see figure below).

What is the maximum length of *AB*?

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Source: K.R.S. Kastry
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The solution will be posted shortly.
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