Problem: Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. The ratio of the area of the first circle to the sum of areas of all other circles in the sequence, is
Answer Choices:
A. (4+32β):4
B. 92β:2
C. (16+122β):1
D. (2+22β):1
E. (3+22β):1
Solution:
Let O denote the vertex of the right angle, C and Cβ² the centers, r and rβ² ( r>rβ² ) the radii of any two consecutive circles. If T is the point of contact of the circles, then OT=OCβ²+rβ²=(2β+1)rβ² and OT=OCβr= ( 2ββ1 )r. Equating these expressions for OT yields the ratio of consecutive radii rβ²/r=(2ββ1)/(2β+1)=(2ββ1)2. If r is the radius of the first circle in the sequence, then
Οr2 is its area, and the sum of the areas of all the other circles, which form a geometric series, is
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