Find the equation in cartesian coordinates of the locus of z if |z + 8

1.	Find the equation in cartesian coordinates of the locus of z if \|z + 8\| = \|z – 4\|
Answer» Let z = x + iy \|z + 8\| = \|z – 4\| \|x + iy + 8\| = \|x + iy – 4\| \|(x + 8) + iy \| = \|(x – 4) + iy\| \(\sqrt{(x+8)^2+y^2}=\sqrt{(x-4)^2+y^2}\) (x + 8)² + y² = (x - 4)² + y² x² + 16x + 64 + y² = x² – 8x + 16 + y² 16x + 64 = -8x + 16 24x + 48 = 0 ∴ x + 2 = 0

Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|

Answer»

Let z = x + iy

|z + 8| = |z – 4|

|x + iy + 8| = |x + iy – 4|

|(x + 8) + iy | = |(x – 4) + iy|

\(\sqrt{(x+8)^2+y^2}=\sqrt{(x-4)^2+y^2}\)

(x + 8)² + y² = (x - 4)² + y²

x² + 16x + 64 + y² = x² – 8x + 16 + y²

16x + 64 = -8x + 16

24x + 48 = 0

∴ x + 2 = 0