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}#