Question

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

Answer 1

(x+y)/3=(y+z)/4  ==>4x+4y=3y+3z ==>y=3z-4x  (i)

(x+y)/3=(z+x)/5 ==> 5x+5y=3z+3x ==> 5y=3z-2x   (ii)

(i)-(ii)

-4y=-2x ==> x=2y

Usando (i)  ==> y=3z-8y ==> 9y=3z  ==>z=3y

(xy+yz+zx)/(x²+y²+z²)

= (2y²+3y²+6y²)/(4y²+y²+9y²)

=11y²/(14y²) =11/14  é a resposta