# 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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