Brilliant Family

Geometry Level 5

Let \(P\) be any point on the line \(x-y+3=0\) and \(A\) be a fixed point \((3,4)\). If the family of lines given by the equation \((3\sec \theta + 5\csc \theta)x+(7\sec \theta-3\csc \theta)y+11(\sec \theta - \csc \theta)=0\) are concurrent at a point \(B\) for all permissible values of \(\theta\) and the maximum value of \(|PA-PB|=2\sqrt{n}\), where\(n\) is a square-free positive integer, then find the value of \(n\).

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