# 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 $S$.

×