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If z₁, z₂ and z₃, z₄ are 2 pairs of complex conjugate numbers, then $arg\left ( \frac{Z_{1}}{Z_{4}} \right )+arg\left ( \frac{Z_{2}}{Z_{3}} \right )$ equals:

Option 1)
$0$

Option 2)
$\frac{\pi }{2}$

Option 3)
$\frac{3\pi }{2}$

Option 4)
$\pi$

Answers (2)

best_answer

As we have learned

Conjugate of a Complex Number -

$z=a+ib \Rightarrow \bar{z}=a-ib$

- wherein

$\bar{z}$ denotes conjugate of z

Definition of Argument/Amplitude of z in Complex Numbers -

$\theta =tan^{-1}|\frac{y}{x}|, z\neq 0$

$\boldsymbol{\theta,\pi-\theta,-\pi+\theta,-\theta}$ are Principal Argument if z lies in first, second, third or fourth quadrant respectively.

- wherein

$z_1= \bar{z_2}$

$z_3= \bar{z_4}$

So, $arg\left ( \frac{z_1}{z_4} \right )+ arg\left ( \frac{z_2}{z_3} \right )$

$arg\left ( \frac{z_1}{ \bar z_3} \right )+ arg\left ( \frac{\bar z_1}{z_3} \right )$

$arg \overline{\left ( \frac{\bar z_1}{z_3 } \right )}+arg \left ( \frac{\bar z_1}{z_3} \right )$

$=-arg {\left ( \frac{\bar z_1}{z_3 } \right )}+arg \left ( \frac{\bar z_1}{z_3} \right )$

$=0$

Option 1)

$0$

Option 2)

$\frac{\pi }{2}$

Option 3)

$\frac{3\pi }{2}$

Option 4)

$\pi$

Posted by

Himanshu

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