Question

a and b are two positive integers such that a+b=a/b+b/a find the value of a^2+b^2

Answer 1

a+b=a/b+b/a

-> Now take L.C.M

a+b=(a^2+b^2)/ab

-> Cross multiplying

(a+b)ab=a^2+b^2

a^2*b+b^2*a=a^2+b^2

a^2*b+b^2*a -a^2-b^2 =0

Take a^2 and b^2common

a^2(b−1)+b^2(a−1)=0------------------(1)

Now since a and b are positive integers - their square can’t be zero.

So in order to make the equation 1 equal to zero:

(b−1) and (a−1) both has to be 0.

Therefore,

b−1=0 =>b=1

and,

a−1=0 =>a=1

So a^2 +b^2 =1^2+1^2 =1+1

=2