Question

If a, b and c are all non-zero and a + b + c = 0, then prove that $\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=3$.

Answer 1

To prove, $\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=3$,
We know that, $a^3+b^3+c^3–3abc=(a+b+c)(a^2+b^2+c^2–ab–bc–ca)$
$=0(a^2+b^2+c^2–ab–bc–ca)$ $[\because a+b+c=0,given]$
$=0$
$\to a^3+b^3+c^3=3abc$
On dividing both sides by abc; we get,
$\frac{a^3}{abc}+\frac{b^3}{abc}+\frac{c^3}{abc}=3$
$\Rightarrow \frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=3$