Let G be the geometric mean of two positive numbers a and b , and M be the arithmetic mean of 1 a and 1 ~b . If 1 M : G is 4: 5 , then a : b can be:

Step-by-step solution

G = a b aligned & M = 1 a + 1 b 2 & M = a+b 2 a b aligned Given that 1 M : G =4: 5 aligned & 2 a b (a+b) a b = 4 5 & a+b 2 a b = 5 4 & a+b+2 a b a+b-2 a b = 5+4 5-4 aligned Using Componendo & Dividendo aligned & ( a )^2+( b )^2+2 a b ( a )^2+( b )^2-2 a b = 9 1 & ( b + a )^2 b - a )^2= 9 1 b + a b - a = 3 1 & b + a + b - a b + a - b + a = 3+1 3-1 aligned Using Componendo & Dividendo b a = 4 2 =2 aligned b a &= 4 1 a b &= 1 4 a: b=1: 4 aligned