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

Answer: (B)

Let $z=x+iy$
$\therefore \frac{2z+1}{iz+1}=\frac{2(x+iy)+1}{i(x+iy)+1}=\frac{2x+1+iy}{(-y+1)+ix}$
$=\frac{[(2x+1)+iy][(-y+1)-ix]}{[(-y+1)+ix][(-y+1)-ix]}$
$=\frac{(2x+1)(-y+1)-(2x+1)xi+y(-y+1)j+xy}{(-y+1{)}^2+x^2}$
$=\frac{(2x+1)(-y+1)+xy+\left(-y^2+y-2x^2-x\right)i}{(-y+1{)}^2+x^2}$
It is given that imaginary part of $\frac{2z+1}{iz+1}=-2$
$\therefore \frac{-y^2+y-2x^2-x}{(-y+1{)}^2+x^2}=-2$
$\Rightarrow -y^2+y-2x^2-x=-2(-y+1{)}^2-2x^2$
$\Rightarrow -y^2+y-x=-2y^2-2+4y$
$\Rightarrow y^2-3y-x+2=0$
which represent the equations of parabola.