Question

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