First slide

Geometry of complex numbers

Question

Perimeter of the locus represented by $\arg (\frac{z + i}{(z - i)}) = \frac{π}{4}$ , (where $i = \sqrt{- 1}$ ) is equal to:

Moderate

A

$\frac{3 π}{2}$
B

$\frac{3 π}{\sqrt{2}}$
C

$\frac{π}{\sqrt{2}}$
D

None of these

Solution

$∵$                       $∠ Q P R = π / 4$
$∴ ∠ Q C R = π / 2$ , where $C$ center of circle.
Now in $Δ Q C R$ .
               $r^{2} + r^{2} = {(Q R)}^{2} = {| 2 i |}^{2} = 4$
$∴ r = \sqrt{2}$
$∴$ Required perimeter $= 2 π r - \frac{1}{4} (2 π r)$
            $= \frac{3}{2} π r = \frac{3}{2} π (\sqrt{2}) = \frac{3 π}{\sqrt{2}}$ .

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