First slide

Geometry of complex numbers

Question

The equation $|\frac{(1 + i) z - 2}{(1 + i) z + 4}| = k$ does not represent a circle when k is

Moderate

A

2
B

$π$
C

$e$
D

1

Solution

We can write the equation as
$|\frac{z - \frac{2}{(1 + i)}}{z + \frac{4}{(1 + i)}}| = k$ (1)
But $\frac{2}{1 + i} = \frac{1^{2} - i^{2}}{1 + i} = 1 - i$
and $\frac{4}{1 + i} = 2 (1 - i)$
$|\frac{z - (1 - i)}{z + 2 (1 - i)}| = k$ (2)
This will not represent a circle if $k = 1$ . When $k = 1, (2)$ represents perpendicular bisector of the segment joining $- 2 (1 - i) a n d 1 - i .$

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