Question

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

Answer 1

$$\begin{gathered} \mathrm{Given}:a+b+c=0 \\ \mathrm{As}\mathrm{we}\mathrm{know}\mathrm{that}, \\ \mathrm{if}a+b+c=0,\mathrm{then}{a}^3+b^3+c^3=3abc \\ \mathrm{Therefore},a^3+b^3+c^3=3abc \\ \mathrm{Dividing}\mathrm{by}\left(abc\right)\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow \frac{a^3+b^3+c^3}{abc}=\frac{3abc}{abc} \\ \Rightarrow \frac{a^3}{abc}+\frac{b^3}{abc}+\frac{c^3}{abc}=3 \\ \Rightarrow \frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=3 \\ \mathrm{Hence},\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=3. \end{gathered}$$