Pushing the boundaries

Geometry Level 5

Let circle AA have the equation x2+(y2)2=4x^{2} + (y - 2)^{2} = 4 and circle BB have the equation x2+(y+1)2=9x^{2} + (y + 1)^{2} = 9.

Now let C1C_{1} be the circle in the first quadrant that is tangent to A,BA, B and the yy-axis, and C2C_{2} be the circle in the first quadrant that is tangent to A,BA, B and the xx-axis.

If the center of C1C_{1} is at (x1,y1)(x_{1}, y_{1}) and the center of C2C_{2} is at (x2,y2)(x_{2}, y_{2}), then find

(x2y2)(x1+3y1)(\dfrac{x_{2}}{y_{2}})*(x_{1} + 3y_{1}).

Clarification: To guarantee a unique solution, both C1C_{1} and C2C_{2} must lie inside BB and must both lie entirely within the first quadrant, (inclusive of the xx and yy axes).

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