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# Series of Circles

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## Series of Circles

Triangle ABC is placed on a complex plane so that the modulus of A is 0, B = 12 + 0i and C can be represented in the form C = k + i
81 - k2
, for some value of k. The area of the triangle is
162
5
.

Circle C1 lies in the first quadrant, and it is tangent to line AB and line AC. The radius of C1 is 1.

For each circle Cn (n > 1), circle Cn is also tangent to lines AB and AC, is externally tangent to circle Cn - 1, and its center is closer to point A than the center of circle Cn - 1.

Ellipse E has a major axis of length 4, and its area is equal to the sum of the areas of C1, C2, C3, ...

Find the eccentricity of ellipse E to the nearest 100th of a unit.

Note: This problem was designed for my honors students who have studied trigonometry, complex numbers, coordinate geometry and conics, and sequences and series this year. This problem incorporates elements of all of those areas of study.

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Problem by Mr. Twitchell

## Solution

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