Question:
If $z_1$ and $z_2$ are two non-zero complex numbers such that $| z_1+ z_2| = | z_1| + | z_2|$ , then $arg (z_1 ) - arg (z_2)$ is equal to
If $z_1$ and $z_2$ are two non-zero complex numbers such that $| z_1+ z_2| = | z_1| + | z_2|$ , then $arg (z_1 ) - arg (z_2)$ is equal to
Updated On: Jul 28, 2022
- $-\pi$
- $-\frac{\pi}{2}$
- 0
- $\frac{\pi}{2}$
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The Correct Option is C
Solution and Explanation
Given, $|z_1+z_2|=|z_1|+|z_2|$
On squaring both sides, we get
$|z_1|^2+|z_2|^2+2|z_1|z_2|| \, \cos \, (arg \, z_1 -arg \, z_2)$
$\hspace20mm \, =|z_1|^2+|z_2|^2+2|z_1||z_2|$
$\Rightarrow \, \, \, 2|z_1||z_2| \, \cos \, (arg z_1-arg \, z_2)=2|z_1||z_2|$
$\Rightarrow \, \, \, \, \cos(arg \, z_1 \, - \, arg \, z_2)=1$
$\Rightarrow \, \, \, arg(z_1)-arg(z_2) =0$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
