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Question
Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|
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Solution
Let z = x + iy, then
|z + 8| = |z – 4| gives
|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
∴ 24x + 48 = 0
∴ x + 2 = 0
This is the equation of the required locus.
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