Question

If A is the A.M. between a and b, prove that :
(i) (A - a)² + (A - b)²=1/2 (a - b)²
(ii) 4 (a - A) (A - b) = (a - b)².​

Answer 1

Answer:

The idea is to just expand the formulae.

We know

(a+b)2=a2+b2+2ab

(a−b)2=a2+b2−2ab

So,

12[(a+b)2+(a−b)2]

=12[(a2+b2+2ab)+(a2+b2−2ab)]

Cancelling +2ab and −2ab and adding like terms a2 and b2 ,

=12[2a2+2b2]

Taking 2 as common from a2 and b2 ,

=12[2(a2+b2)]

Cancelling the 2,

=a2+b2

Another method:

From LHS,

a2+b2=a2+b2−2ab+2ab

a2+b2=(a−b)2+2ab⋯(1)

Also

a2+b2=a2+b2+2ab−2ab

a2+b2=(a+b)2−2ab⋯(2)

Add equations (1) and (2),

(a2+b2)+(a2+b2)=((a−b)2+2ab)+((a+b)2−2ab)

⟹2a2+2b2=(a−b)2+(a+b)2

⟹2(a2+b2)=(a−b)2+(a+b)2

⟹(a2+b2)=12[(a−b)2+(a+b)2]