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