Question

If x = a + b, y = b + c, z = c + a, where

a + b + c is not equal to 0 and a, b, c are not equal, then the value of

$\frac{ {x}^{3} + {y}^{3} + {z}^{3} - 3xyz }{{a}^{3} + {b}^{3} + {c}^{3} - 3abc }$
is:​

Answer 1

Given system of equations can be written as AX=0

where A=

⎣

⎢

⎢

⎡

1

−c

−b

−c

1

−a

−b

−a

1

⎦

⎥

⎥

⎤

,X=

⎣

⎢

⎢

⎡

0

0

0

⎦

⎥

⎥

⎤

,O=

⎣

⎢

⎢

⎡

0

0

0

⎦

⎥

⎥

⎤

For non-trivial solution,

∣A∣=0

∣

∣

∣

∣

∣

∣

∣

∣

1

−c

−b

−c

1

−a

−b

−a

1

∣

∣

∣

∣

∣

∣

∣

∣

=0

⇒1−a

2

−b

2

−c

2

−2abc=0

⇒a

2

+b

2

+c

2

+2abc=1