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

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

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

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

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

answer is A, B, C.

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Detailed Solution

$Option (a) ∵ PS ∥ QR$
$\begin{array}{ll} ∴ \arg (\frac{z_{1} - z_{4}}{z_{2} - z_{3}}) = 0 \\ \Rightarrow \frac{z_{1} - z_{4}}{z_{2} - z_{3}} is purely real. \end{array}$
Option (b) $∵$ Diagonals of rhombus are perpendicular.
Then, $\arg (\frac{z_{1} - z_{3}}{z_{2} - z_{4}}) = \frac{π}{2}$
$\Rightarrow \frac{z_{1} - z_{3}}{z_{2} - z_{4}}$ is purely imaginary.
Option (c ) $∵$ $PR \neq QS$
$\begin{array}{llr} ∴ |z_{1} - z_{3}| \neq |z_{2} - z_{4}| \\ Option (d) ∵ ∠ QSP = ∠ RSQ \end{array}$
$\begin{array}{l} ∴ amp (\frac{z_{2} - z_{4}}{z_{1} - z_{4}}) = amp (\frac{z_{3} - z_{4}}{z_{2} - z_{4}}) \\ \Rightarrow - amp (\frac{z_{1} - z_{4}}{z_{2} - z_{4}}) = - amp (\frac{z_{2} - z_{4}}{z_{3} - z_{4}}) \\ \Rightarrow amp (\frac{z_{1} - z_{4}}{z_{2} - z_{4}}) = amp (\frac{z_{2} - z_{4}}{z_{3} - z_{4}}) \end{array}$