Question

If $$ax = by = cz$$, prove that $$\frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} = \frac{bc}{a^2} + \frac{ca}{b^2} + \frac{ab}{c^2}$$.

\[ [\textbf{Hint :}\ ax = by = cz,\ \Rightarrow \ \dfrac{x}{bc} = \dfrac{y}{ca} = \dfrac{z}{ab} = k \Rightarrow x = kbc,\ y = kca \ \textit{and} \ z = kab.] \]

Answer 1

Given: $$ax = by = cz$$

To prove: $$\frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} = \frac{bc}{a^2} + \frac{ca}{b^2} + \frac{ab}{c^2}$$

Proof:

1. Dividing $$ax = by = cz$$ by $$abc$$, we get $$\frac{x}{bc} = \frac{y}{ca} = \frac{z}{ab} = k$$.
2. Then $$x = kbc$$, $$y = kca$$, $$z = kab$$.
3. $$\frac{x^2}{yz} = \frac{(kbc)^2}{(kca)(kab)} = \frac{k^2 b^2 c^2}{k^2 a^2 bc} = \frac{bc}{a^2}$$
4. $$\frac{y^2}{zx} = \frac{(kca)^2}{(kab)(kbc)} = \frac{k^2 c^2 a^2}{k^2 ab^2 c} = \frac{ca}{b^2}$$
5. $$\frac{z^2}{xy} = \frac{(kab)^2}{(kbc)(kca)} = \frac{k^2 a^2 b^2}{k^2 abc^2} = \frac{ab}{c^2}$$
6. $$\text{L.H.S.} = \frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} = \frac{bc}{a^2} + \frac{ca}{b^2} + \frac{ab}{c^2} = \text{R.H.S.}$$

Hence proved.