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`.

Answer 1

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.