Solveeit Logo
Login

Question

Mathematics Question on Complex Numbers and Quadratic Equations

If z1z_1 and z2z_2 are two non-zero complex numbers such that ∣z1+z2∣=∣z1∣+∣z2∣| z_1+ z_2| = | z_1| + | z_2| , then arg(z1)−arg(z2)arg (z_1 ) - arg (z_2) is equal to

A

−π-\pi

B

−π2-\frac{\pi}{2}

C

0

D

π2\frac{\pi}{2}

Answer

0

Explanation

Solution

Given, ∣z1+z2∣=∣z1∣+∣z2∣|z_1+z_2|=|z_1|+|z_2| On squaring both sides, we get ∣z1∣2+∣z2∣2+2∣z1∣z2∣∣ cos⁡ (arg z1−arg z2)|z_1|^2+|z_2|^2+2|z_1|z_2|| \, \cos \, (arg \, z_1 -arg \, z_2) \hspace20mm \, =|z_1|^2+|z_2|^2+2|z_1||z_2| ⇒   2∣z1∣∣z2∣ cos⁡ (argz1−arg z2)=2∣z1∣∣z2∣\Rightarrow \, \, \, 2|z_1||z_2| \, \cos \, (arg z_1-arg \, z_2)=2|z_1||z_2| ⇒    cos⁡(arg z1 − arg z2)=1\Rightarrow \, \, \, \, \cos(arg \, z_1 \, - \, arg \, z_2)=1 ⇒   arg(z1)−arg(z2)=0\Rightarrow \, \, \, arg(z_1)-arg(z_2) =0