Q. 84.4( 7 Votes )

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio (3 + 2) : (3 – 2).

Answer :

Let the two numbers be a and b.


GM = √ab


According to the given condition,


a + b = 6√ab …(1)


(a + b)2 = 36ab


Also,


(a – b)2 = (a + b)2 – 4ab


= 36ab – 4ab


= 32ab


a–b = √32ab


= 4√2ab …..(2)


Adding (1) and (2), we obtain


2a = (6 + 4√2)√ab


a = (3 + 2√2)√ab


substituting the value of a in (1), we obtain,


b =(3 – 2√2)√ab



Thus, the required ratio is (3+2√2) : 3–2√2.


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