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If $z=x+i y$, then the equation $|z+1|=|z-1|$ represents
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Verified Answer
The correct answer is:
$Y$-axis
Given,
$\begin{gathered}
z=x+i y \\
|z+1|=|z-1| \\
|z+1|^{2}=|z-1|^{2} \\
|x+i y+1|^{2}=|x+i y-1|^{2} \\
(x+1)^{2}+y^{2}=(x-1)^{2}+y^{2} \\
(x+1)^{2}-(x-1)^{2}=0 \\
(x+1-x+1)(x+1+x-1)=0 \\
x=0 \\
\therefore|z+1|=|z-1| \text { represents } Y \text {-axis. }
\end{gathered}$
$\begin{gathered}
z=x+i y \\
|z+1|=|z-1| \\
|z+1|^{2}=|z-1|^{2} \\
|x+i y+1|^{2}=|x+i y-1|^{2} \\
(x+1)^{2}+y^{2}=(x-1)^{2}+y^{2} \\
(x+1)^{2}-(x-1)^{2}=0 \\
(x+1-x+1)(x+1+x-1)=0 \\
x=0 \\
\therefore|z+1|=|z-1| \text { represents } Y \text {-axis. }
\end{gathered}$
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