The ratio of the A.M and G.M of two positive numbers a and b, is m : n. Show that a : b = (m + √m² - n²) : (m - √m² - n²)

Solution:

Let the two numbers be a and b.

A.M = (a + b)/2

G.M = √ab

According to the given condition,

⇒ (a + b) / (2√ab) = m/n

Squaring on both sides,

⇒ (a + b)²/4ab = m²/n²

⇒ (a + b)² = 4abm²/n²

⇒ (a + b) = 2m√ab/n ....(1)

Using this in the identity (a - b)² = (a + b)² - 4ab, we obtain

⇒ (a - b)² = 4abm²/n² - 4ab

⇒ (a - b)² = 4ab (m² - n²)/n²

⇒ (a - b) = 2√ab √m² - n²/n

Adding (1) and (2), we obtain

2a = (2√ab)/n (m + √m² - n²)

a = (√ab)/n (m + √m² - n²)

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

b = (2√ab)/n x m - (√ab)/n x (m + √m² - n²)

= (√ab)/n x m - (√ab)/n x (√m² - n²)

= (√ab)/n (m - √m² - n²)

Therefore,

a/b = [(√ab)/n (m + √m² - n²)] / [(√ab)/n (m - √m² - n²)]

= (m + √m² - n²) / (m - √m² - n²)

Thus, a : b = (m + √m² - n²) : (m - √m² - n²)

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NCERT Solutions Class 11 Maths Chapter 9 Exercise ME Question 19

The ratio of the A.M and G.M of two positive numbers a and b , is m : n. Show that a : b = (m + √m² - n²) : (m - √m² - n²)

Summary:

It was known that the ratio of A.M and G.M of two positive numbers a and b is m:n we showed that a : b = (m + √m² - n²) : (m - √m² - n²)