Circle squared

Calculus Level 3

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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