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Question

Two equal circles of radius $$R$$ are touching each other externally. If a smaller circle of radius $$r$$ is touching both of these circles as well as their direct common tangent, then the ratio $$r : R$$ is


A
1:2
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B
1:2
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C
1:22
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D
1:4
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Solution

The correct option is D $$1 : 4$$

Length of common tangent to circles of equal radii = $$2 \times radius$$

Here $$l = 2R$$

$$\implies AD = DC = BE = \dfrac{1}{2} = R$$

$$AB = R + r$$

$$AE = R - r$$

$$AB^2 = AE^2 + BE^2$$

$$\implies (R + r)^2 = (R - r)^2 + R^2$$

$$\implies 4Rr = R^2$$

$$\implies \dfrac{4}{1} = \dfrac{R}{r} \implies r:R = 1:4$$


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