If $\mathrm{P}(x, y)$ denotes $\mathrm{z}=x+\mathrm{i} y x, y \in \mathbb{R}$ and $\mathrm{i}=\sqrt{-1}$ in…
If $\mathrm{P}(x, y)$ denotes $\mathrm{z}=x+\mathrm{i} y x, y \in \mathbb{R}$ and $\mathrm{i}=\sqrt{-1}$ in Argand's plane and $\left|\frac{z-1}{z+2 i}\right|=1$, then the locus of $P$ is
parabola
hyperbola
circle
straight line
Solution
$\begin{aligned} & \left|\frac{\mathrm{z}-1}{\mathrm{z}+2 \mathrm{i}}\right|=1 \\ & \Rightarrow\left|\frac{x+\mathrm{i} y-1}{x+\mathrm{i} y+2 \mathrm{i}}\right|=1 \\ & \Rightarrow|x-1+\mathrm{i} y|=|x+(y+2) \mathrm{i}| \\ & \Rightarrow \sqrt{(x-1)^2+y^2}=\sqrt{x^2+(y+2)^2} \\ & \Rightarrow x^2-2 x+1+y^2=x^2+y^2+4 y+4 \\ & \Rightarrow 2 x+4 y+3=0, \text { which is a straight line }\end{aligned}$