Question

if z=x+iy is a complex number such that modulus of z =modulus of z-i then the locus of z is

Answer 1

Given a complex number $z=x+iy$ such that,

$\longrightarrow|z|=|z-i|$

$\longrightarrow|x+iy|=|x+iy-i|$

$\longrightarrow|x+iy|=|x+i(y-1)|$

Taking the modulus,

$\longrightarrow\sqrt{x^2+y^2}=\sqrt{x^2+(y-1)^2}$

Squaring,

$\longrightarrow x^2+y^2=x^2+(y-1)^2$

Subtracting $x^2,$

$\longrightarrow y^2=(y-1)^2$

$\longrightarrow y^2=y^2-2y+1$

Subtracting $y^2,$

$\longrightarrow0=-2y+1$

$\longrightarrow2y-1=0$

$\longrightarrow\underline{\underline{y=\dfrac{1}{2}}}$

Hence the locus of $z$ is a straight line parallel to x axis and passing through the point (0, 1/2).