If $x, y, z$ are not all zero and if the equations $a x + b y + c z = 0, b x + c y + a z = 0. c x + a y + b z = 0$ then $x^{3}$ : $y^{3}$ : $z^{3} =$

1 : 2 : 3

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1 : 1 : 1

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1 : 3 : 2

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2 : 3 : 1

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Solution

The correct option is B $1 : 1 : 1$
$a x + b y + c z = 0$
$b x + c y + a z = 0$
$+ a x + a y + b z = 0$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (a + b + c) x + (a + b + c) y + (a + b + c) z = 0$
$(x + y + z) (a + b + c) = 0 x + y + z = 0$
$\Rightarrow x = y = z b e c a u s e x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) \cdot \frac{1}{2} [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]$

$\Rightarrow x^{3} : y^{3} : z^{3} = 1 : 1 : 1$

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