Single Choice

If $$z_1$$, $$z_2$$are conjugate complex numbers and arg $$\left ( z_1z_2 \right )=$$

A$$arg z_1+arg z_2=0$$
Correct Answer
B$$arg z_1 + argz_2+n\pi$$, $$n\epsilon \left \{ -1,0,1 \right \}$$
C$$arg z_1 + argz_2+n\pi$$, $$n\epsilon N$$
D$$arg z_1-argz_2$$

Solution

Let $$z_{1}=r_{1}e^{i\theta _{1}}$$
$$z_{2}=r_{2}e^{i\theta _{2}}=\overline{z_1}=r_{1}e^{-i\theta _{1}}$$
$$z_1z_2=r_1r_2$$
$$\therefore arg(z_1z_2)=0$$ or $$\pm \pi$$ $$(\because$$ $$z_1z_2)$$ is purely real.
$$arg(z_1)+arg(z_2)=\theta_1-\theta_1=0$$
Hence, option A is correct.

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