Question

If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$, prove that $$\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}$$.

Answer 1

Given: $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$

To prove: $$\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}$$

Proof:

1. Let $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c} = k$$, then $$x = ka, y = kb, z = kc$$
2. $$\text{L.H.S.} = \frac{(ka)^3}{a^3} + \frac{(kb)^3}{b^3} + \frac{(kc)^3}{c^3} = k^3 + k^3 + k^3 = 3k^3$$
3. $$\text{R.H.S.} = \frac{3xyz}{abc} = \frac{3(ka)(kb)(kc)}{abc} = \frac{3k^3 abc}{abc} = 3k^3$$
4. $$\text{L.H.S.} = \text{R.H.S.}$$

Hence proved.