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
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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