Question

If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$, prove that $$\frac{x^3}{a^2} + \frac{y^3}{b^2} + \frac{z^3}{c^2} = \frac{(x + y + z)^3}{(a + b + c)^2}$$.

Answer 1

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

To prove: $$\frac{x^3}{a^2} + \frac{y^3}{b^2} + \frac{z^3}{c^2} = \frac{(x + y + z)^3}{(a + b + c)^2}$$

Proof:

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

Hence proved.