Question

If the imaginary part of $\frac{2z+1}{iz+1}$ is $-2,$ then the locus of $z$ is

• A. a circle
• B. a straight line
• C. an ellipse
• D. a parabola

Answer 1

The correct option is B a straight line

Let $z=x+iy$
$\frac{2z+1}{iz+1}=\frac{2(x+iy)+1}{i(x+iy)+1}$
$=\frac{(2x+1)+2iy}{(1-y)+ix}\times \frac{(1-y)-ix}{(1-y)-ix}$
$=\frac{(2x+1)+2iy}{(1-y{)}^2+x^2}\times \left\{\right(1-y)-ix\}$
$\therefore$ Imaginary part
$=\frac{2y(1-y)-x(2x+1)}{(1-y{)}^2+x^2}=-2$
$\Rightarrow 2y-2y^2-2x^2-x=-2x^2-2+4y-2y^2$
$\Rightarrow x+2y-2=0$ which is a straight line.