Question

The ratio of the A.M and G.M. of two positive numbers a and b , is m : n . Show that .

Answer 1

The given ratio of A.M. and G.M. of two positive number a and b is m:n.

We know that, A.M.= a+b 2 and G.M.= ab .

It can be concluded from the given condition that,

a+b 2 ab = m n ( a+b ) 2 4( ab ) = m 2 n 2 ( a+b ) 2 = 4ab m 2 n 2 ( a+b )= 2 ab m n (1)

Relation between a and b can be given by,

( a−b ) 2 = ( a+b ) 2 −4ab

Substitute the value of ( a+b ) 2 in the above relation,

( a−b ) 2 = 4ab m 2 n 2 −4ab ( a−b ) 2 = 4ab( m 2 − n 2 ) n 2 ( a−b )= 2 ab m 2 − n 2 n (2)

Add equation (1) and (2),

2a= 2 ab n ( m+ m 2 − n 2 ) a= ab n ( m+ m 2 − n 2 )

Substitute value of a in (1) to determine b,

b= 2 ab n m− ab n ( m+ m 2 − n 2 ) = ab n ( m− m 2 − n 2 )

Divide a and b,

a b = ab n ( m+ m 2 − n 2 ) ab n ( m− m 2 − n 2 ) = ( m+ m 2 − n 2 ) ( m− m 2 − n 2 )

Hence, it is proved that a:b=( m+ m 2 − n 2 ):( m− m 2 − n 2 ).