A light source at the point $$(0, 16)$$ in the coordinate plane casts light in all directions. A disc (a circle along with its interior) of radius 2 with center at $$(6, 10)$$ casts a shadow on the $$X$$ axis. The length of the shadow can be written in the form $$m\sqrt{n}$$ where $$m, n$$ are positive integers and $$n$$ is square-free. Find $$m+n$$.

A line $$y = 16 + mx$$ through the source touches the disc when $$\frac{|6m+6|}{\sqrt{m^2+1}} = 2$$, which reduces to $$4m^2 + 9m + 4 = 0$$. The two slopes satisfy $$m_1m_2 = 1$$ and $$m_1 - m_2 = \frac{\sqrt{17}}{4}$$, and each tangent meets the $$X$$ axis at $$x = -\frac{16}{m}$$. The shadow length is $$16 \cdot \frac{|m_1-m_2|}{|m_1m_2|} = 4\sqrt{17}$$, so $$m+n = 4+17 = 21$$.