First slide

Geometry of complex numbers

Question

$Im (\frac{2 z + 1}{i z + 1}) = 5$ represents

Moderate

A

$a$ circle
B

$a$ straight line
C

$a$ parabola
D

$a n$ ellipse

Solution

$\frac{1}{2 i} [\frac{2 z + 1}{i z + 1} - \frac{2 \bar{z} + 1}{- i \bar{z} + 1}] = 5$
$\begin{array}{l} \Rightarrow (2 z + 1) (i \bar{z} - 1) + (2 \bar{z} + 1) (i z + 1) \\ = 10 i (i z + 1) (i \bar{z} - 1) \\ \Rightarrow 4 i z \bar{z} + (- 2 + i) z + (2 + i) \bar{z} \\ = 10 i (- z \bar{z} - i z + i \bar{z} - 1) \\ \Rightarrow 14 i z \bar{z} + (8 + i) z + (- 8 + i) \bar{z} - 10 i = 0 \\ \Rightarrow z \bar{z} + \frac{1}{14} (1 - 8 i) z + \frac{1}{14} (1 + 8 i) \bar{z} - \frac{5}{7} = 0 \end{array}$
This represents a circle

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