HOW TO FIND EQUATION OF LOCUS OF COMPLEX NUMBERS

Question :

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

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.