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Question
If a, b, c are all non-zero and a + b + c = 0, prove that `a^2/(bc) + b^2/(ca) + c^2/(ab) = 3`.
Solution
To prove, `a^2/(bc) + b^2/(ca) + c^2/(ab) = 3`
We know that, a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
= 0(a2 + b2 + c2 – ab – bc – ca) ...[∵ a + b + c = 0, given]
= 0
⇒ a3 + b3 + c3 = 3abc
On dividing both sides by abc, we get
`a^3/(abc) + b^3/(abc) + c^3/(abc) = 3`
⇒ `a^2/(bc) + b^2/(ac) + c^2/(ab) = 3`
Hence proved.
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