Question

# The complex number $$z$$ satisfying the equation $$|z - i| = |z + 1| = 1$$ is

A
0
B
1+i
C
1+i
D
1i

Solution

## The correct options are B $$-1 + i$$ C $$0$$Given : $$|z-i|=|z+1|=1$$Let $$z=x+iy$$$$|x+iy-i|=1$$$$\implies x^{2}+(y-1)^{2}=1$$$$\implies x^{2}+y^{2}-2y+1=1$$$$x^{2}+y^{2}-2y=0$$ .... $$(i)$$Also, $$|z+1|=1$$$$\implies |x+iy+1|=1$$$$\implies (x+1)^{2}+y^{2}=1$$$$\implies x^{2}+y^{2}+2x=0$$ ..... $$(ii)$$Equating $$(i)$$ and $$(ii)$$ we get$$-y=x$$For $$y=1, x=-1$$$$\therefore z=-1+i$$Also, $$y=0\implies x=0$$$$\therefore z=0$$Mathematics

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