Question

Find the locus of z which satisfies the condition |i+z| / |i-z| =1​

Answer 1

Given,

$\longrightarrow\dfrac{|i+z|}{|i-z|}=1$

Take $z=x+iy$ and,

$\longrightarrow\dfrac{|i+(x+iy)|}{|i-(x+iy)|}=1$

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

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

Taking the modulus,

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

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

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

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

$\longrightarrow2y=-2y$

$\longrightarrow4y=0$

$\longrightarrow\underline{\underline{y=0}}$

This implies locus of $z$ is x axis.