# Series of Circles

Pro Problems > Math## 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 - k

, for some value of k. The area of the triangle is ^{2}162

5

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

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

Ellipse E has a major axis of length 4, and its area is equal to the sum of the areas of C_{1}, C_{2}, C_{3}, ...

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.

Presentation mode

Problem by Mr. Twitchell

## Solution

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