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