# 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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