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+32​):4
B. 92​:2
C. (16+122​):1
D. (2+22​):1
E. (3+22​):1
Solution:
Let O denote the vertex of the right angle, C and C′ the centers, r and r′ ( r>r′ ) the radii of any two consecutive circles. If T is the point of contact of the circles, then OT=OC′+r′=(2​+1)r′ and OT=OC−r= ( 2​−1 )r. Equating these expressions for OT yields the ratio of consecutive radii r′/r=(2​−1)/(2​+1)=(2​−1)2. If r is the radius of the first circle in the sequence, then
Ï€r2 is its area, and the sum of the areas of all the other circles, which form a geometric series, is
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