How do you identify the locus in the complex plane given by: |z - 3i| = |z +2|?

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Answer 1 · 2015-12-19T08:08:17

${x + (-2/3x + 5/6)i: x in RR}$

Let $z = x + iy$ where $x, y in RR$

$|z-3i| = |z+2|$

$=> |x+iy - 3i| = |x + iy + 2|$

$=> |x + (y-3)i| = |(x+2) + iy|$

$=> sqrt(x^2 + (y-3)^2) = sqrt((x+2)^2 + y^2)$

$=> x^2 + (y-3)^2 = (x+2)^2 + y^2$

$=> x^2 + y^2 -6y + 9 = x^2 + 4x + 4 + y^2$

$=> y = -2/3x + 5/6$

$=> z = x + (-2/3x + 5/6)i$

As we placed no restrictions on $x$ beyond $x in RR$ , this gives us the result

${z in CC : |z-3i| = |z+2|} = {x + (-2/3x + 5/6)i: x in RR}$