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Evaluation of a Determinant

If 1 / a+1 / ...

Question

If 1/a + 1/b + 1/c = 1/(a+b+c), where a+b+c and abc is not equal to zero then what's the value of

(a+b)(b+c)(c+a)

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Solution

1/a + 1/b + 1/c =1/ (a+b+c)
=> 1/a + 1/b =1/ (a+b+c) - 1/c
=> (b+a)/ab = (c - (a+b+c)) / c(a+b+c)

Cross multiplying (Since, a,b,c !=0 and a+b+c != 0)

=> c(a+b+c)(b+a) = -ab(a+b)
=>(a+b)( ca +cb +cc + ab ) = 0
=> (a+b)( ca + cc + cb + ab) = 0
=> (a+b)(b+c)(c+a)=0

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