Circles

Geometry Level 5

\[x^{2}+y^{2}+3\sqrt{2}(x+y)=0 \\ x^{2}+y^{2}+5\sqrt{2}(x+y)=0\]

Consider the equations of the two circles above. A third circle is drawn internally tangent to the larger circle, externally tangent to the smaller circle, and tangent to their common diameter. If the third circle's radius can be written as \(\frac{2p-1}{p}\), then what is the value of \(p\)?

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