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

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$a d + b c = 0$

$a d - b c = 0$

$a b + c d = 0$

$a b - c d = 0$

answer is B.

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Detailed Solution

Since, $z_{1}, z_{2}$ and $z_{1} - z_{2}$ are collinear

$∴ |\begin{array}{ccc} z_{1} & {\bar{z}}_{1} & 1 \\ z_{2} & {\bar{z}}_{2} & 1 \\ z_{1} - z_{2} & \bar{z_{1} - z_{2}} & 1 \end{array}| = 0$

$\Rightarrow |\begin{array}{ccc} z_{1} & {\bar{z}}_{1} & 1 \\ z_{2} & {\bar{z}}_{2} & 1 \\ z_{1} - z_{2} & {\bar{z}}_{1} - {\bar{z}}_{2} & 1 \end{array}| = 0$

Applying $R_{3} \to R_{3} - R_{1} + R_{2}, then |\begin{array}{ccc} z_{1} & {\bar{z}}_{1} & 1 \\ z_{2} & {\bar{z}}_{2} & 1 \\ 0 & 0 & 1 \end{array}| = 0$

$\begin{array}{l} z_{1} {\bar{z}}_{2} - {\bar{z}}_{1} z_{2} = 0 \\ \Rightarrow z_{1} {\bar{z}}_{2} - ({\overset{}{z}}_{1} {\bar{z}}_{2}) = 0 \\ \Rightarrow Im (z_{1} {\bar{z}}_{2}) = 0 \\ \Rightarrow \\ \Rightarrow Im ((a + i b) (c - i d)) = 0 \\ \Rightarrow b c - a d = 0 \Rightarrow a d - b c = 0 \end{array}$