Circle squared

Calculus Level 4

Consider a square \(ABCD\). Two circles \(X\) and \( Y\) are drawn such that circle \(X\) is tangent to two adjacent sides \(AB\) and \(AD\), and circle \(Y\) is tangent to the other two adjacent sides \(CB \) and \(CD,\) and circles \(X\) and \(Y\) are tangent to each other.

As the radius of circle \(X\) increases, at what radius of circle \(X\) does the sum of the areas of circle \(X\) and \(Y\) change the slowest?

Express that radius as a fraction of the side of the square \(ABCD\), and round to the nearest hundredth.

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