$P (z_{1}), Q (z_{2}), R (z_{3})$ and $S (z_{4})$ are four complex numbers representing the vertices of a rhombus taken in order on the complex plane, then which one of the following is/are correct ?

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$\frac{z_{1} - z_{4}}{z_{2} - z_{3}}$ is purely real

amp $\frac{z_{1} - z_{4}}{z_{2} - z_{4}} \neq$ amp $\frac{z_{2} - z_{4}}{z_{3} - z_{1}}$

$\frac{z_{1} - z_{3}}{z_{2} - z_{4}}$ is purely imaginary

$| z_{1} - z_{3} | \neq | z_{2} - z_{4} |$

answer is A, B, C, D.

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Detailed Solution

A $P S | | Q R \Rightarrow \arg (\frac{z_{1} - z_{4}}{z_{2} - z_{3}}) = 0 \Rightarrow \frac{z_{1} - z_{4}}{z_{2} - z_{3}}$ is purely real

B $a m p (\frac{z_{1} - z_{4}}{z_{2} - z_{4}}) \neq a m p (\frac{z_{2} - z_{4}}{z_{3} - z_{1}})$

C. Diagonals of rhombs are perpendicular. Hence, $\frac{z_{1} - z_{3}}{z_{2} - z_{4}}$ is purely imaginary

D. Diagonals of rhombus are not equal. Hence, $| z_{1} - z_{3} | \neq | z_{2} - z_{4} |$ .

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