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In the Argand plane, the values of $\mathrm{Z}$ satisfying the equation $|\mathrm{z}-1|=|\mathrm{i}(\mathrm{z}+1)|$ lie on
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Verified Answer
The correct answer is:
the $\mathrm{Y}$ - axis
$$
\begin{aligned}
& \text { }|\mathrm{z}-1|=|\mathrm{i}(\mathrm{z}+1)| \\
& \Rightarrow|\mathrm{z}-1|=|\mathrm{i}| \cdot|\mathrm{z}+1| \\
& \Rightarrow|\mathrm{z}-1|^2=|\mathrm{z}+1|^2 \\
& \Rightarrow|\mathrm{x}+\mathrm{iy}-1|^2=|\mathrm{x}+\mathrm{iy}+1|^2 \\
& \Rightarrow(\mathrm{x}-1)^2+\mathrm{y}^2=(\mathrm{x}+1)^2+\mathrm{y}^2 \\
& \Rightarrow \mathrm{x}^2+1-2 \mathrm{x}=\mathrm{x}^2+1+2 \mathrm{x} \\
& \Rightarrow 4 \mathrm{x}=0 \Rightarrow \mathrm{x}=0
\end{aligned}
$$
which is equation of the $y$-axis.
\begin{aligned}
& \text { }|\mathrm{z}-1|=|\mathrm{i}(\mathrm{z}+1)| \\
& \Rightarrow|\mathrm{z}-1|=|\mathrm{i}| \cdot|\mathrm{z}+1| \\
& \Rightarrow|\mathrm{z}-1|^2=|\mathrm{z}+1|^2 \\
& \Rightarrow|\mathrm{x}+\mathrm{iy}-1|^2=|\mathrm{x}+\mathrm{iy}+1|^2 \\
& \Rightarrow(\mathrm{x}-1)^2+\mathrm{y}^2=(\mathrm{x}+1)^2+\mathrm{y}^2 \\
& \Rightarrow \mathrm{x}^2+1-2 \mathrm{x}=\mathrm{x}^2+1+2 \mathrm{x} \\
& \Rightarrow 4 \mathrm{x}=0 \Rightarrow \mathrm{x}=0
\end{aligned}
$$
which is equation of the $y$-axis.
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