Question

Determine o valor de
$\frac{xy + yz + zx}{x ^2 + y ^2 + z ^{2} }$
para
$\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$
​

Answer 1

$\mathsf{\frac{x}{3} = \frac{y}{4} = \frac{z}{5}=k} \\\mathsf{x = 3k \: y = 4k \: z = 5k}$

$\mathsf{\frac{xy + yz + zx}{x ^2 + y ^2 + z ^{2}} = \frac{3k.4k + 4k.5k + 5k.3k}{ {(3k)}^{2} + {(4k)}^{2} + {(5k)}^{2} }}$

$\mathsf{\dfrac{12{k}^{2}+20{k}^{2}+15{k}^{2}}{9{k}^{2}+16{k}^{2}+25{k}^{2}}=\dfrac{47 {k}^{2} }{50 {k}^{2} } = \dfrac{47}{50} }$