Question

What is your answer: 1/a+1/b=1/a+b, Help me, a=?, b=?

Answer 1

1/a + 1/b = 1/(a + b)

( a + b) /ab = 1/( a + b)

(a + b)^2 = ab

a^2 + b^2 +2ab = ab

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

we know ,
a^3 - b^3 =(a - b)(a^2 + b^2 + ab)
put equation (1)
a^3 - b^3 = (a - b)(0)= 0
a^3 = b^3
take cube root
a = b

now,
1/a + 1/b =2/a
but given,
1/a + 1/b =1(a+b)
because both are opposite nature.
so, This isn't possible .

Answer 2

1/a  +  1/b  =  1/(a+b)
(a+b) / ab  =  1/(a+b)

(a+b)² = ab
a² + b² + 2 ab = ab
a² + b² = - a b
We know that  always  a² + b² >= 2 a b            as (a-b)² >= 0
               but  - a b < 2 a b

Hence there is no solution to the given problem.
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Another  way:

a² + b² = - ab

As LHS is always positive, a*b is negative.  Let us assume that b is negative.
   let b = - c   , where c is positive.    and  a is positive.

So   a² + c²  =  a c 
        a² + c² - 2 a c = - a c
        (a - c)² = - a c

This is not possible.  There is no solution to the given problem. There are no real a , b such that    1/a  + 1/b  = 1/(a+b).