The sum of two numbers is 6 times their G.M, show that numbers are in the ratio (3 + 2√2) : (3 - 2√2)

Solution:

Let the two numbers be a and b.

Then its G.M. = √ ab

According to the given condition,

⇒ a + b = 6√ab ....(1)

⇒ (a + b)² = 36(ab)

Also,

(a - b)² = (a + b)² - 4ab

= 36ab - 4ab

= 32ab

a - b = √32 √ ab

= 4√ 2 √ ab ...(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 = 6√ab - (3 + 2√2) √ ab

= (3 - 2√2) √ ab

Hence the ratio of the numbers is

a/b = [(3 + 2√2)√ ab] / [(3 - 2√2) √ ab]

= (3 + 2√2) / (3 - 2√2)

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

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NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.3 Question 28

The sum of two numbers is 6 times their G.M, show that numbers are in the ratio (3 + 2√2) : (3 - 2√2)

Summary:

We were given that the sum of two numbers is 6 times their G.M, we showed that the numbers are in the ratio (3 + 2√2) : (3 - 2√2)