Two circles touch externally. The sum of their areas is 41\pi square cm. If the distance between their centres is 9 cm, then what is difference between their diameters?

Explanation

Let radii be r_1 and r_2. Given r_1+r_2 = 9 and \pi r_1^2 + \pi r_2^2 = 41\pi \implies r_1^2 + r_2^2 = 41. Using (r_1+r_2)^2 = r_1^2 + r_2^2 + 2r_1r_2 \implies 81 = 41 + 2r_1r_2 \implies 2r_1r_2 = 40. Difference (r_1-r_2)^2 = (r_1+r_2)^2 - 4r_1r_2 = 81 - 80 = 1 \implies r_1-r_2 = 1. The difference between diameters is 2(r_1-r_2) = 2(1) = 2\text{ cm}.

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