If x, y, z are not all zero and if ax + by + cz = 0, bx + cy + az = 0 and cx + ay + bz = 0, then x : y : z may be

1 : 1 : 1

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1 : w : w^{2}

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1 : w^{2} : w

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w : 1 : w^{2}

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Solution

The correct options are
A $1 : 1 : 1$
B $1 : w : w^{2}$
C $1 : w^{2} : w$
D $w : 1 : w^{2}$
$D = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ \Rightarrow 3 a b c - a^{3} - b^{3} - c^{3} = 0 \Rightarrow (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = 0 \Rightarrow (a + b + c) (a + b w + c w^{2}) (a + b w^{2} + c w) = 0 \Rightarrow a + b + c = 0 o r a + b w + c w^{2} = 0 o r a + b w^{2} + c w = 0 \Rightarrow c l e a r l y x : y : z = 1 : 1 : 1 o r 1 : w : w^{2} o r 1 : w^{2} : w 1 : w^{2} : w (o r w : 1 : w^{2})$

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