1. Let $$C$$ be a circle with centre $$P_0$$ and $$AB$$ be a diameter of $$C$$. Suppose $$P_1$$ is the mid point of the line segment $$P_0B$$,$$P_2$$ is the mid point of the line segment $$P_1B$$ and so on. Let $$C_1,C_2,C_3,...$$ be circles with diameters $$P_0P_1, P_1P_2, P_2P_3...$$ respectively. Suppose the circles $$C_1, C_2, C_3,...$$ are all shaded. The ratio of the area of the unshaded portion of $$C$$ to that of the original circle is





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  • By: anil on 05 May 2019 02.30 am
    Radius of the circles $$C_1, C_2, C_3,...$$ would be in GP with (R/4),(R/8),(R/16) and so on. Radii of circles are in the ratio 1:4.  Ratio of unshaded region to the ratio of original circle = 1-$$frac{Ratio of shaded region}{Ratio of original circle}$$ = 1-$$frac{pi r^2/16 + pi r^2/64+.....}{pi r^2}$$ = 1-$$frac{1/16}{(1-1/4)}$$ = 1- 1/12 = $$frac{11}{12}$$ = 11:12
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