First slide

Geometry of complex numbers

Question

If the imaginary part of $\frac{2 z + 1}{i z + 1}$ is $- 4,$ then the locus of the point representing z in the complex plane is

Moderate

A

a straight line
B

a parabola
C

a circle
D

an ellipse

Solution

Let $z = x + i y$ then
$\begin{array}{l} \frac{2 z + 1}{i z + 1} = \frac{2 (x + i y) + 1}{i (x + i y) + 1} = \frac{(2 x + 1) + 2 i y}{(1 - y) + i x} \\ = \frac{[(2 x + 1) + 2 i y] [(1 - y) - i x]}{(1 - y)^{2} + x^{2}} \end{array}$
As $Im (\frac{2 z + 1}{i z + 1}) = - 4$ , we get
$\frac{2 y (1 - y) - x (2 x + 1)}{x^{2} + (1 - y)^{2}} = - 4$
$\begin{array}{ll} \Rightarrow 2 x^{2} + 2 y^{2} + x - 2 y = 4 x^{2} + 4 (y^{2} - 2 y + 1) \\ \Rightarrow 2 x^{2} + 2 y^{2} - x - 6 y + 4 = 0 \end{array}$
which represents a circle.

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