Question

If $P(x, y)$ denotes $z = x + i y $ in Argand's plane and $| \frac {z-1}{z+2i}| = 1$, then the locus of $P$ is a/an

• A. hyperbola
• B. ellipse
• C. circle
• D. straight line

Answer 1

Answer: (D)

Given, $z=x+i y$ and $\left|\frac{z-1}{z+2 i}\right|=1$
$\Rightarrow \left|\frac{(x+i y)-1}{(x+i y)+2 i}\right|=1$
$\Rightarrow |(x-1)+i y|=|x+(y+2) i|$
Squaring on both sides,
$|(x-1)+i y|^{2}=|x+(y+2) i|^{2}$
$\Rightarrow (x-1)^{2}+y^{2}=x^{2}+(y+2)^{2}$
$\Rightarrow x^{2}+y^{2}+1-2 x=x^{2}+y^{2}+4+4 y$
$\Rightarrow 2 x+24+3=0$, which represents a straight line.