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Question

If ab+bc+ca=0 then prove that 1a2bc+1b2ca+1c2ab=0.

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Solution

Given,
ab+bc+ca=0.....(1).
Now,1a2bc+1b2ca+1c2ab
=1a2+ab+ac+1b2+ab+bc+1c2+ac+bc [ Since ab=ac+bc,bc=ab+ac,ca=ab+bc using (1)]
=1(a+b+c)(1a+1b+1c)
=1(a+b+c)(ab+bc+caabc)
=0 [Using (1)]

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