Question

If $S=\{z\in C:\frac{z-i}{z+2i}\in R\}$, then

• A. $S$ contains exactly two elements
• B. $S$ is a circle in the complex plane
• C. $S$ is a straight line in the complex plane
• D. $S$ contains only one element

Answer 1

The correct option is C $S$ is a straight line in the complex plane

Given : $\frac{z-i}{z+2i}$
Let $z=x+iy$
$\Rightarrow \frac{x+(y-1)i}{x+(y+2)i}\times \frac{x-(y+2)i}{x-(y+2)i}$
$\Rightarrow \frac{x^2+(y-1)(y+2)+i\left(x\right(y-1)-x(y+2\left)\right)}{x^2+(y+2{)}^2}$
Since $\frac{z-i}{z+2i}$ is real
$\therefore \frac{\left(x\right(y-1)-x(y+2\left)\right)}{x^2+(y+2{)}^2}=0$
$\Rightarrow x(y-1)-x(y+2)=0$
$\Rightarrow -x-2x=0$
$\therefore x=0$, which represents the imaginary axis.