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Question
Find the equation in cartesian coordinates of the locus of z if `|("z" + 3"i")/("z" - 6"i")|` = 1
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Solution
Let z = x + iy, then
`|("z" + 3"i")/("z" - 6"i")|` = 1 gives
`|(x + iy + 3"i")/(x + iy - 6"i")|` = 1
∴ `|(x + (y + 3)"i")/(x + (y - 6)"i")| = 1 ...[because |"z"_1/"z"_2| = |"z"_1/"z"_2|]`
∴ |x + (y + 3)i = |x + (y – 6)i|
∴ `sqrt(x^2 + (y + 3)^2) = sqrt(x^2 + (y - 6)^2)`
x2 + (y + 3)2 = x2 + (y – 6)2
∴ x2 + y2 + 6y + 9 = x2 + y2 – 12y + 36
∴ 18y – 27 = 0
∴ 2y – 3 = 0
This is the equation of the required locus.
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