Question

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

• A. A circle
• B. A parabola
• C. A straight line
• D. An ellipse

Answer 1

The correct option is C A straight line

Given that imaginary part of $\frac{2z+1}{iz+1}$ is $-2$.

Hence, $\frac{2(x+iy)+1}{i(x+iy)+1}=\frac{(2x+1)+2i\left(y\right)}{(1-y)+ix}=\frac{\left[\right(2x+1)+2iy]\left[\right(1-y)-ix]}{{(1-y)}^2+x^2}$
$\text{Im}=-2$
Thus $-2x^2-x+2y-2y^2=[-x^2-y^2+1](-2)$ is a straight line.