Question

If ${{Z}_{1}}$ and ${{Z}_{2}}$ two non-zero complex numbers such that $\left| {{Z}_{1}} \right|+\left| {{Z}_{2}} \right|=\left| {{Z}_{1}} \right|\left| {{Z}_{2}} \right|$ then $\arg \left( {{Z}_{1}}/{{Z}_{2}} \right)$ is equal to
$\left( 1 \right)\text{ 0}$
$\left( 2 \right)\text{ }-\pi$
$\left( 3 \right)\text{ }-\dfrac{\pi }{2}$
$\left( 4 \right)\text{ }\dfrac{\pi }{2}$
$\left( 5 \right)\text{ }\pi$

Answer 1

In this question we have been given with two non-zero complex numbers which are ${{Z}_{1}}$ and ${{Z}_{2}}$. We have been given with the data that the modulus of the sum of the two complex numbers is equal to the product of the modulus of the two complex numbers. We will solve this question by using the property of parallel complex numbers and get the required solution for the same.

Complete step by step answer:
We know from the question that:
$\Rightarrow \left| {{Z}_{1}} \right|+\left| {{Z}_{2}} \right|=\left| {{Z}_{1}} \right|\left| {{Z}_{2}} \right|$
Where ${{Z}_{1}}$ and ${{Z}_{2}}$ are two non-zero complex numbers.
We have to find the value of $\arg \left( {{Z}_{1}}/{{Z}_{2}} \right)$.
We know the argument of a complex number is the angle to which the exponent is raised on the complex number when it is written in the exponential form.
Consider the value of ${{Z}_{1}}={{r}_{1}}{{e}^{i{{\theta }_{1}}}}$ and ${{Z}_{2}}={{r}_{2}}{{e}^{i{{\theta }_{2}}}}$.
We can see from the given numbers that $\arg {{Z}_{1}}={{\theta }_{1}}$ and $\arg {{Z}_{2}}={{\theta }_{2}}$
Now on dividing the two complex numbers, we get:
$\Rightarrow \dfrac{{{Z}_{1}}}{{{Z}_{2}}}=\dfrac{{{r}_{1}}{{e}^{i{{\theta }_{1}}}}}{{{r}_{2}}{{e}^{i{{\theta }_{2}}}}}$
Now using the property of division of exponential complex numbers, we get:
$\Rightarrow \dfrac{{{Z}_{1}}}{{{Z}_{2}}}=\dfrac{{{r}_{1}}}{{{r}_{2}}}{{e}^{i\left( {{\theta }_{1}}-{{\theta }_{2}} \right)}}$
Therefore, we can conclude that:
$\Rightarrow \arg \dfrac{{{Z}_{1}}}{{{Z}_{2}}}={{\theta }_{1}}-{{\theta }_{2}}$
Now we have been given the data that $\left| {{Z}_{1}} \right|+\left| {{Z}_{2}} \right|=\left| {{Z}_{1}} \right|\left| {{Z}_{2}} \right|$ since the addition and the product is the same, this implies that the lines of the complex numbers are parallel which implies the angle between them is $0$. Therefore, we can write:
$\Rightarrow \arg \dfrac{{{Z}_{1}}}{{{Z}_{2}}}={{\theta }_{1}}-{{\theta }_{2}}=0$, which is the required solution.

So, the correct answer is “Option 1”.

Note: The various forms of writing a complex number should be remembered. A complex number can be written as $z=x+iy$, in the polar form it can be written as $r\left( \cos \theta +i\sin \theta \right)$ and in the exponential form it can be written as $z=r{{e}^{i\theta }}$, where $\theta$ is the argument function, which is denoted as $\arg \left( z \right)$, where $z$ denotes the complex number.