Gently does it

Geometry Level 4

A circle of radius \(r\) is positioned so that it has two points of tangency with the parabola \(y = x^{2}\). Another circle, again of the same radius \(r\), is positioned (entirely in the first quadrant) so that it is tangent to both \(y = x^{2}\) and the positive \(x\)-axis.

There is a unique value of \(r\) so that the two circles described above, both of radius \(r\), are also tangent to one another. If \(r = \dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers, then find \(a + b\).

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