Question

the sum of two numbers is 6 times their Geometric mean, show that numbers are in ratio (3+2√2):(3-2√2)​

Answer 1

Answer:

let the numbers be a and b

geometric mean of a and b is given by √ab

given,

a+b=6√ab

(a+b)/2√ab = 3/1

using componendo and dividendo

(a+b+2√ab)/(a+b-2√ab) = (3+1)/(3-1)

(√a + √b)^2 / √a - √b)^2 = 4/2 = 2

(√a + √b) / (√a - √b) = √2

using componendo and dividendo again

(√a+√b + √a-√b)/(√a+√b - √a + √b) = (√2+1)/(√2-1)

2√a/2√b = (√2+1)/(√2-1)

√a/√b = (√2+1)/(√2-1)

squaring both side

a/b = (2+1+2√2)/(2+1-2√2)

a/b = (3+2√2)/(3-2√2)

proved