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If the points represented by complex numbers $z_{1} = a + i b, z_{2} = a^{1} + i b^{1} and z_{1} - z_{2}$ are collinear, then

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$a b^{1} + a^{1} b = 0$

$a b + a^{1} b^{1} = 0$

$a b - a^{1} b^{1} = 0$

$a b^{1} - a^{1} b = 0$

answer is B.

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Detailed Solution

By definition, $z_{1} = x_{1} + i y_{1}; z_{2} = x_{2} + i y_{2}; z_{3} = x_{3} + i y_{3}$ are collinear, if

$|\begin{array}{lll} x_{1} y_{1} 1 \\ x_{2} y_{2} 1 \\ x_{3} y_{3} 1 \end{array}| = 0 \Rightarrow |\begin{array}{ccc} a & b & 1 \\ a^{'} & b^{'} & 1 \\ a - a^{'} & b - b^{'} & 1 \end{array}| = 0$

$\Rightarrow a b^{'} = a^{'} b$

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