First slide

Geometry of complex numbers

Question

Perimeter of the locus represented by $A r g (\frac{z + i}{z - i}) = \frac{π}{4}$ is equal to ____( where $i = \sqrt{- 1}$ )

Moderate

A

$\frac{6 π}{\sqrt{2}}$
B

$\frac{3 π}{\sqrt{2}}$
C

$\frac{3 π}{2}$
D

$2 \sqrt{2} π$

Solution

$∵ Q P R = \frac{π}{4}$
$∴ Q C R = \frac{π}{2}$ Where ‘c’ is centre of circle
Locus is the major segment of the circle with centre at C and radius $\sqrt{2}$ $(i n ∆ Q C R, r^{2} + r^{2} = {|2 i|}^{2} \Rightarrow r = \sqrt{2})$
Perimeter $= 2 π r - \frac{π r}{2}$
$= \frac{3}{2} π r = \frac{3 π}{\sqrt{2}}$

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