The equation a r g z - 1 z + 1 = π 4 represents a circle with:

We have, a r g z - 1 z + 1 = π 4 a r g z 0 - 1 z 0 + 1 = π 2 ⇒ z 0 - 1 z 0 + 1 = e i π 2 ⇒ z 0 - 1 z 0 + 1 = i ⇒ z 0 = 1 + i 1 - i ⇒ z 0 = ( 1 + i ) 2 2 ⇒ z 0 = i ⇒ Centre 0 , 1 Radius is the distance between z 0 and - 1 , 0 = - 1 - 0 2 + 0 - 1 2 = 2 units

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Mathematics - Complex Number Question with Solution | TestHub

MathematicsComplex NumberGeometric ApplicationMedium2 minPYQ_2021

MathematicsMediumsingle choice

The equation $a r g (\frac{z - 1}{z + 1}) = \frac{π}{4}$ represents a circle with:

Options:

Answer:

Solution:

We have,

$a r g (\frac{z - 1}{z + 1}) = \frac{π}{4}$

$a r g (\frac{z_{0} - 1}{z_{0} + 1}) = \frac{π}{2}$

$\Rightarrow (\frac{z_{0} - 1}{z_{0} + 1}) = e^{i \frac{π}{2}}$

$\Rightarrow \frac{z_{0} - 1}{z_{0} + 1} = i$

$\Rightarrow z_{0} = (\frac{1 + i}{1 - i})$

$\Rightarrow z_{0} = \frac{(1 + i)^{2}}{2}$

$\Rightarrow z_{0} = i$

$\Rightarrow$ Centre $(0, 1)$

Radius is the distance between $z_{0}$ and $(- 1, 0)$

$= \sqrt{{(- 1 - 0)}^{2} + {(0 - 1)}^{2}}$

$= \sqrt{2} units$

Stream:JEESubject:MathematicsTopic:Complex NumberSubtopic:Geometric Application

⏱ 2mℹ️ Source: PYQ_2021

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