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If Z1, Z2 and Z3, Z4 Are Two Pairs of Conjugate Complex Numbers, Prove that Arg ( Z 1 Z 4 ) + Arg ( Z 2 Z 3 ) = 0 . - Mathematics

If z1z2 and z3z4 are two pairs of conjugate complex numbers, prove that \[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\].

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Solution

Given that z1z2 and z3z4 are two pairs of conjugate complex numbers.

\[\therefore z_1 = r_1 e^{i \theta_1} , z_2 = r_1 e^{- i \theta_1} , z_3 = r_2 e^{i \theta_2} \text { and } z_4 = r_2 e^{- i \theta_2}\]

Then,

\[\frac{z_1}{z_4} = \frac{r_1 e^{i \theta_1}}{r_2 e^{- i \theta_2}} = \frac{r_1}{r_2} e^{i\left( \theta_1 - \theta_2 \right)} \]

\[ \Rightarrow \arg\left( \frac{z_1}{z_4} \right) = \theta_1 - \theta_2 . . . (1)\]

and

\[\frac{z_2}{z_3} = \frac{r_1 e^{- i \theta_1}}{r_2 e^{i \theta_2}} = \frac{r_1}{r_2} e^{i\left( - \theta_1 + \theta_2 \right)} \]

\[ \Rightarrow \arg\left( \frac{z_2}{z_3} \right) = \theta_2 - \theta_1 . . . (2)\]

\[\therefore \arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = \theta_1 - \theta_2 - \theta_1 + \theta_2 \]

        \[ = 0\]

Hence,  

\[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.4 | Q 5 | Page 57
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