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Question

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


A
0
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B
1+i
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C
1+i
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D
1i
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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, $$y=0\implies x=0$$
$$\therefore z=0$$

673341_635725_ans_2755f43ff5064313acfb6472a55b7984.png

Mathematics

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