if a2+b2=5ab prove that a3/b3+b3/a3=110 - Maths -

Question

if a2+b2=5ab prove that
a3/b3+b3/a3=110

Manbar Singh · Accepted Answer

We have,a2 + b2 = 5 ab [given] (1)We have to prove that, a3b3 +
b3a3 = 110

LHS = a3b3 + b3a3=a6 + b6a3b3=a23 + b23a3b3=a2 + b2a22 + b22 -
a2b2a3b3 as, x3 + y3 = (x + y) (x2 + y2 - xy)=a2 + b2a4 + b4 -
a2b2a3b3=a2 + b2a4 + b4 - a2b2 + 2 a2b2 - 2 a2b2a3b3=a2 + b2a4 + b4
+ 2a2b2 - 3a2b2 a3b3=a2 + b2a22 + b22 + 2a2b2- 3a2b2 a3b3=a2 + b2a2
+ b22 - 3a2b2a3b3=5ab5ab2 - 3a2b2a3b3 using (1)

= 5ab25a2b2 - 3a2b2a3b3=5ab × 22a2b2a3b3=110a3b3a3b3 = 110 =
RHS