# Find the equation in cartesian coordinates of the locus of z if |z+3iz-6i| = 1 - Mathematics and Statistics

Sum

Find the equation in cartesian coordinates of the locus of z if |("z" + 3"i")/("z" - 6"i")| = 1

#### 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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