Question

The ratio of A M and G. M of two positive no. a and b is m : n
show that
a : b = { m + √( m^2- n^2)} : { m - √( m^2 - n^2)​

Answer 1

Step-by-step explanation:

AM of number a and b = (a + b)/2

GM of number a and b = √ab

Here,

AM : GM = m : n

(a + b)/2 : √ab = m : n

(a + b)/2√ab = m/n

Applying componendo and dividendo rule,

(a + b + 2√ab)/(a + b - 2√ab) = (m + n)/(m - n)

(√a² + √b² + 2√ab)/(√a² + √b² -2√ab) = (m + n)/(m - n)

(√a + √b)²/(√a - √b)² = (m + n)/(m - n)

take square root both sides,

( √a + √b )/(√a - √b) = √(m + n)/√(m - n)

again applying componendo and dividendo,

( √a + √b + √a - √b)/(√a + √b - √a + √b) = {√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }

2√a/2√b = {√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }

√a/√b = {√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }

taking square both sides,

a/b =[{√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }]²

a/b = {m + n + m - n - 2√(m² - n²)}/{m + n + m - n -2√(m² - n²)}

a/b = {2m + 2√(m² - n²)}/{2m - 2√(m² - n²)}

a/b = {m + √(m² - n²)}/{m - √(m² - n²)}

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

Hope it helps

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