Question

find the locus if a complex number is z=x+yi satisfying the relation |z+i|=|z+2|

Answer 1

Given,
$|z + i| = |z + 2|$
$|x + iy + i| = |x + iy + 2|$
$|x + (y + 1)i| = |(x + 2) + iy|$
$\sqrt{ {x}^{2} + {(y + 1)}^{2} } = \sqrt{ {(x + 2)}^{2} } + {y}^{2}$
${x}^{2} + {y }^{2} + 1 + 2y = {x}^{2} + 4 + 4x + {y}^{2}$
1 + 2y = 4 + 4x
4x - 2y + 3 = 0
It is the required equation of locus.