The sum of the areas of two circles, which touch each other externally, is 153π. If the sum of their radii is 15, find the ratio of the larger to the smaller radius.
Explanation:
If the radii of two circles are r1 and r2, then the two equations can be written πr12 + πr22 = 153π or (r12 + r22) = 153 and r1 + r2 = 15. Now r12 + r22 = (r1 + r2)2 – 2r1r2 Therefore, 153 = (15)2 – 2r1r2 or r1r2 = 36. If 36 is to be expressed as the product of two integers, it could be (36 × 1), (18 × 2), (12 × 3), (9 × 4), (6 × 6). The only two factors that add up to 15 are 12 and 3.
Hence, r1 = 12, r2 = 3. Therefore, the ratio of larger radius to the smaller one is 12 : 3 = 4.
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