# HOW TO FIND EQUATION OF LOCUS OF COMPLEX NUMBERS

How to Find Equation of Locus of Complex Numbers ?

Here we are going to see some example problems based on how to find equation of locus of complex numbers

## How to Find Equation of Locus of Complex Numbers

Question 1 :

Obtain the Cartesian equation for the locus of z = x + iy in each of the following cases:

(i) |z − 4| = 16

Solution :

z = x + iy

|(x + iy) − 4| = 16

|(x  - 4) + iy| = 16

√(x - 4)2 + y2  =  16

Taking squares on both sides, we get

(x - 4)2 + y2  =  256

x2 - 2x(4) + 42 + y2  =  256

x2 + y2 - 8x + 16 - 256  =  0

x2 + y2 - 8x - 240  =  0

(ii) |z − 4|2 - |z - 1|2 = 16

Solution :

z = x + iy

|z − 4|2 - |z - 1|2 = 16

|(x + iy) − 4|2 - |(x + iy) - 1|2 = 16

|(x - 4) + iy|2 - |(x - 1) + iy|2 = 16

(√(x - 4)2 + y2)2 - (√(x - 1)2 + y2)2  =  16

(x - 4)2 + y2 - [(x - 1)2 + y2]  =  16

x2 - 8x + 16 + y2[x2 - 2x + 1 + y2]  =  16

x2 - 8x + 16 + y2 - x2 + 2x - 1 - y2  =  16

-6x + 15 - 16  =  0

-6x - 1  =  0

Multiply through out the equation by negative, we get

6x + 1  =  0

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