If {z_1} and {z_2} are two non-zero complex n | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given that $|{z_1} + {z_2}| = |{z_1}| + |{z_2}|$. This condition implies that the complex numbers ${z_1}$ and ${z_2}$ lie on the same ray originat

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) Given that $|{z_1} + {z_2}| = |{z_1}| + |{z_2}|$. This condition implies that the complex numbers ${z_1}$ and ${z_2}$ lie on the same ray originating from the origin in the complex plane. Let ${z_1} = {r_1}(\cos{	heta_1} + i\sin{	heta_1})$ and ${z_2} = {r_2}(\cos{	heta_2} + i\sin{	heta_2})$. Then $|{z_1} + {z_2}|^2 = (|{z_1}| + |{z_2}|)^2 = |{z_1}|^2 + |{z_2}|^2 + 2|{z_1}||{z_2}|$. Also,$|{z_1} + {z_2}|^2 = (z_1 + z_2)(\overline{z_1} + \overline{z_2}) = |z_1|^2 + |z_2|^2 + z_1\overline{z_2} + \overline{z_1}z_2 = |z_1|^2 + |z_2|^2 + 2	ext{Re}(z_1\overline{z_2})$. Comparing these,$2	ext{Re}(z_1\overline{z_2}) = 2|z_1||z_2|$,which means $\cos(	heta_1 - 	heta_2) = 1$. Thus,$	heta_1 - 	heta_2 = 0$,which implies $	ext{arg}({z_1}) - 	ext{arg}({z_2}) = 0$.

If ${z_1}$ and ${z_2}$ are two non-zero complex numbers such that $|{z_1} + {z_2}| = |{z_1}| + |{z_2}|,$ then $\text{arg}({z_1}) - \text{arg}({z_2})$ is equal to

Similar Questions

The amplitude of $\frac{1 + \sqrt{3}i}{\sqrt{3} + i}$ is

If ${z_1}, {z_2} \in \mathbb{C}$,then $\text{amp}\left( \frac{z_1}{\bar{z}_2} \right) = $

The argument of the complex number $-1 + i\sqrt{3}$ is ............. $^\circ$.

Let $z$ and $w$ be complex numbers such that $\overline{z} + i\overline{w} = 0$ and $\text{arg}(zw) = \pi$. Then $\text{arg}(z)$ equals

Convert the given complex number in polar form: $i$

Vedclass Test Series

Exam Paper Generator

Online Exam Module