Question

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

Answer 1

We have,

$\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=k$ (say)

$x=ka,y=kb,z=kc$

Consider the L.H.S.

$\Rightarrow \frac{x^3}{a^3}+\frac{y^3}{b^3}+\frac{z^3}{c^3}$

$\Rightarrow \frac{k^3{a}^3}{a^3}+\frac{k^3{b}^3}{b^3}+\frac{k^3{c}^3}{c^3}$

$\Rightarrow k^3+k^3+k^3$

$\Rightarrow 3k^3$

Consider the R.H.S.

$\Rightarrow 3\frac{{(x+y+z)}^3}{{(a+b+c)}^3}$

$\Rightarrow 3\frac{{(ka+kb+kc)}^3}{{(a+b+c)}^3}$

$\Rightarrow 3\frac{{k^3(a+b+c)}^3}{{(a+b+c)}^3}$

$\Rightarrow 3k^3$

Hence, $L.H.S=R.H.S$