**How to find locus of complex numbers ?**

To find the locus of given complex number, first we have to replace z by the complex number x + iy and simplify.

Let us look into some example problems to understand the concept.

**Example 1 :**

P represents the variable complex number z, find the locus of P if

Re (z + 1/z + i) = 1

**Solution :**

Let z = x + iy then

By equating the real part of the complex number to 1, we get

[x (x + 1) + y (y + 1)]/x^{2} + (y + 1)^{2 } = 1

(x^{2} + x + y^{2} + y)/x^{2} + (y + 1)^{2 } = 1

(x^{2} + y^{2} + x + y)/(x^{2} + (y^{2} + 2y + 1)) = 1

(x^{2} + y^{2} + x + y)/(x^{2} + y^{2} + 2y + 1) = 1

x^{2} + y^{2} + x + y = x^{2} + y^{2} + 2y + 1

x^{2 }- x^{2} + y^{2} - y^{2} + x + y - 2y - 1 = 0

x - y - 1 = 0

Hence the locus of the given complex number is x - y - 1 = 0.

**Example 2 :**

P represents the variable complex number z, find the locus of P if

|z - 5i| = |z + 5i|

**Solution :**

Let z = x + iy then

|z - 5i| = |z + 5i|

|(x + iy) - 5i| = |(x + iy) + 5i|

|x + i(y - 5)| = |x + i(y + 5)|

√x^{2 }+ (y - 5)^{2 }= √x^{2 }+ (y + 5)^{2}

Taking squares on both sides

x^{2 }+ (y - 5)^{2 }= x^{2 }+ (y + 5)^{2}

x^{2 }+ y^{2} - 10y + 25^{ }= x^{2 }+ y^{2} + 10y+ 25

x^{2 }+ y^{2 - }x^{2 - }y^{2 }- 10y - 10 y + 25 - 25^{ }= 0

-20y = 0

y = 0

Hence the locus of given complex number is y = 0.

**Example 3 :**

P represents the variable complex number z, find the locus of P if

| 2z − 3 | = 2

**Solution :**

Let z = x + iy then

| 2z − 3 | = 2

| 2(x + iy) − 3 | = 2

| 2x + i2y − 3 | = 2

| (2x − 3) + i 2y | = 2

√(2x − 3)^{2} + (2y)^{2} = 2

Taking squares on both sides

(2x)^{2} - 2(2x)(3) + 3^{2} + (2y)^{2} = 2^{2}

4x^{2} - 12x + 9 + 4y^{2} = 4

4x^{2} + 4y^{2}- 12x + 9 - 4 = 0

4x^{2} + 4y^{2}- 12x + 5 = 0

Hence the locus of given complex number is 4x^{2} + 4y^{2}- 12x + 5 = 0.

- Properties of complex numbers
- Add and subtract complex numbers
- How to find the modulus and argument of a complex number

After having gone through the stuff given above, we hope that the students would have understood "How to find locus of complex numbers".

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