Question

Let \[z\] and \[\omega \] be two non zero complex numbers such that \[\left| z \right|=\left| \omega \right|\] and \[Argz+Arg\omega =\pi \], then \[z\] equals
1). \[\omega \]
2). \[-\omega \]
3). \[bar\text{ }\omega \]
4). \[-bar\text{ }\omega \]

Answer 1

Hint: To solve this question try to understand the concept of complex numbers and the properties of complex numbers. You should have the knowledge of arguments of complex numbers and the basic identity of trigonometry i.e. identity of conversion of coordinates. By all these concepts you can easily solve the given question.

Complete step-by-step solution:
To solve this type of question firstly you have to know about the complex numbers and their concepts. And you should also have the knowledge of the concept argument of complex numbers.
Let us first understand the meaning of the complex number. The number which is represented in the form of \[a+ib\], where \[a\] and \[b\] are the real numbers. But the term \[i\] makes the number as imaginary. And it satisfies the equation as \[{{i}^{2}}=-1\]. Complex numbers cannot be represented on the number line.
Argument of a complex number or in short it is denoted as arg, it is defined as the angle between the real positive axis and the line joining the origin and coordinates of \[z\].
Let us solve the question using these concepts.
It is given that \[z\] and \[\omega \] are two non zero complex numbers and as we know that,
\[z=\left| z \right|(\cos \theta +i\sin \theta )\]\[.......(1)\]
Here, \[\theta \] is the argument of the complex number \[z\] or in short it is represented as \[\arg (z)\] .
Let us assume that \[\arg (\omega )={{\theta }_{1}}\] and it is given that \[\arg (z)+\arg (\omega )=\pi \]. We know that, \[\arg (z)=\theta \]
Substituting these values, we will get
\[\theta +{{\theta }_{1}}=\pi \]
Or we can say that, \[\theta =\pi -{{\theta }_{1}}\].
In the question it is given that \[\left| z \right|=\left| \omega \right|\].
Substituting this value in equation \[(1)\], we will get
\[z=\left| \omega \right|(\cos (\pi -{{\theta }_{1}})+i\sin (\pi -{{\theta }_{1}}))\]
But according to the trigonometric identity we know that \[(\pi -{{\theta }_{1}})\] lies in the second coordinate and in the second coordinate the sign of sine function is positive and cosine function is negative. If we apply this trigonometric identity we will get the above equation as,
\[z=\left| \omega \right|(-\cos ({{\theta }_{1}})+i\sin ({{\theta }_{1}}))\]
\[z=-\left| \omega \right|(\cos ({{\theta }_{1}})-i\sin ({{\theta }_{1}}))\]
From the above equation it is clear that the value of \[z\] is the conjugate of \[\omega \]. This means
\[z=-\bar{\omega }\]
Hence we can conclude that option \[(4)\] is correct.

Note: We can apply various operations on complex numbers. We can perform addition, subtraction, multiplication, division on complex numbers. We can also perform conjugation operations on complex numbers i.e. in this operation the sign of the real part remains as it is while the sign of the imaginary part gets changed.