Equation of the locus in complex numbers – Loci problems in complex numbers. Hello friends, today I’ll show how to derive the equation of the locus of any complex number. Have a look!!
The much-needed posts to understand the loci problems are:
Equation of the locus in complex numbers
Solved example of the Equation of the locus in complex numbers
Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.
So here is my example.
According to Stroud and Booth (2013)*, “If , determine the equation of the locus .”
Now here I have to find out the equation of the locus of the complex number . First of all, I’ll get the value of . So I’ll start with the value of .
Since I know that , the value of will be
Now I’ll remove the complex term from the denominatior, that is, the bottom. So I’ll multiply both the top amd bottom with . And that gives
Next, I’ll simplify it. Since , I can say that
As we all know that . So I can rewrite as
Then I’ll simplify it to get
Thus I can separate the real and the complex part of as
Next, I’ll get the equation of the locus.
But from the example itself, I know that
So using equation (1), I can say that
Thus I can say that
And this gives
Now I’ll simplify it. So that gives
So equation (2) is the equation of the locus. And this is the answer to the given example.
Now I’ll give you some further example.
Suppose I have to find out the nature of the locus of the above-mentioned example. And how to do that then?
First of all, I’ll rearrange the terms in equation (2). So it will be
And that means
So, in other words, I can also say that
Thus the locus is a circle with its centre on and radius .
And then this will be the solution to it.
Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!