Question

If $Im\left[\frac{(z-1)}{(2z+1)}\right]=-4$, then locus of $z$ is

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

Answer 1

The correct option is D

a circle

Explanation for correct option:

Find the nature of locus:

Given,

Let $z=x+iy$

$Im\left[\frac{(z-1)}{(2z+1)}\right]=-4$

$\begin{array}{rcl}\frac{(z-1)}{2z+1}& =& \frac{(x+iy-1)}{\left(2\right(x+iy)+1)}\\ & =& \frac{(x-1)+iy}{(2x+1)+2iy}\end{array}$

Multiply numerator and denominator with $(2x+1)-2iy)$

$$\begin{gathered} =\frac{\left(\right(x-1)+iy)\left(\right(2x+1)-2iy)}{\left(\right(2x+1)+2iy)\left(\right(2x+1)-2iy)} \\ =\frac{(x-1)(2x+1)+2y^2+iy(-2x+2+2x+1)}{({(2x+1)}^2+4y^2)} \end{gathered}$$

Further simplifying the above equation,

$$\begin{gathered} Im\left[\frac{(z-1)}{(2z+1)}\right]=-4 \\ \Rightarrow \frac{3y}{\left({(2x+1)}^2+4y^2\right)}=-4 \\ \Rightarrow 3y=-4(4x^2+1+4x+4y^2) \\ \Rightarrow 16x^2+16y^2+16x+3y+4=0 \end{gathered}$$

Hence, this represents the equation of circle.

Hence, the correct option is D.