Connect the curve 3

Calculus Level 3

In this figure, the complete black circle is a circle of equation $${ \left( x-9 \right) }^{ 2 }+{ \left( y-7 \right) }^{ 2 }=2$$

The red line below it is of the equation x=8, with range restrictions $$0<y\le4$$.

If AB touches both the circle and the red straight line and is the shortest of its kind, then its length can be written as $\sqrt { A } \left( \sqrt { B } -C \right)$

If A, B, C are integers with A and B being square free, find A+B+C.

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