Pushing the boundaries

Geometry Level 5

Let circle \(A\) have the equation \(x^{2} + (y - 2)^{2} = 4\) and circle \(B\) have the equation \(x^{2} + (y + 1)^{2} = 9\).

Now let \(C_{1}\) be the circle in the first quadrant that is tangent to \(A, B\) and the \(y\)-axis, and \(C_{2}\) be the circle in the first quadrant that is tangent to \(A, B\) and the \(x\)-axis.

If the center of \(C_{1}\) is at \((x_{1}, y_{1})\) and the center of \(C_{2}\) is at \((x_{2}, y_{2})\), then find

\((\dfrac{x_{2}}{y_{2}})*(x_{1} + 3y_{1})\).

Clarification: To guarantee a unique solution, both \(C_{1}\) and \(C_{2}\) must lie inside \(B\) and must both lie entirely within the first quadrant, (inclusive of the \(x\) and \(y\) axes).

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