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:

*Modulus and argument of the complex numbers*

**Multiplication and division of complex numbers**

**Addition and subtraction of complex numbers**

**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.

**Example **

According to Stroud and Booth (2013)*, “If , determine the equation of the locus .”

**Solution**

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 .

**Step 1**

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

(1)

Next, I’ll get the equation of the locus.

**Step 2**

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

Then I’ll rearrange the terms to get

(2)

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!!

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