Minimal Distance

Geometry Level 4

Consider the circle \(\Gamma\) whose equation is \(x^2 + y^2- 28x + 40y + 20 = 0\). Let \(S\) be the set of all points \(P\) outside \(\Gamma\) such that if there is a line through \(P\) which intersects \(\Gamma\) at two points \(A\) and \(B\), then \(PA\cdot  PB = 100\). Find the minimum possible distance between a point on \(\Gamma\) and a point on \(S\).

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