In the Argand plane, the values of $\mathrm{Z}$ satisfying the equation…

In the Argand plane, the values of $\mathrm{Z}$ satisfying the equation $|\mathrm{z}-1|=|\mathrm{i}(\mathrm{z}+1)|$ lie on
  1. the $\mathrm{Y}$ - axis
  2. a parabola
  3. a hyperbola
  4. the X-axis

Solution

$ \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

Asked in: AP EAMCET 2023 (18 May Shift 2)

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