For any complex number w = c + id , let arg(w) ∈ ( - π, π ] , where i = - 1 . Let α and β be real numbers such that for all complex numbers z = x + iy satisfying arg((z + α)/(z + β)) = (π)/(4) , the ordered pair ( x, y

Correct answer: B. αβ = 4 arg((z + α)/(z + β)) = (π)/(4) represents an arc of a circle and (- α, 0) and (- β, 0) subtends an angle (π)/(4) on z Given, x ^ 2 + y ^ 2 + 5x - 3y + 4 = 0 Centre ((- 5)/(2), (3)/(2)), r = (25)/(4) + (9)/(4) - 4 = 3 2 Angle made by ( - α, 0 ) and ( - β, 0 ) at the centre C is (π)/(2) Using rotation of complex numbers, we get (- α - (- (5)/(2) + (3)/(2)i))/(- β - (- (5)/(2) + (3)/(2)i)) = 1 · e ^ i(π)/(2) = i - α + (5)/(2) - (3)/(2)i = - iβ + (5)/(2)i + (3)/(2) α - iβ = 1 - 4i ∵ α, β ∈ R ⇒ α = 1, β = 4

JEE Advanced 2021Paper 1

Exam: JEE Advanced
Year: 2021
Paper: Paper 1
Subject: Mathematics
Chapter: Complex Number

JEE Advanced 2021 · Paper 1 · Mathematics · Complex Number

For any complex number w = c + id , let arg(w) ∈ ( - π, π ] , where...

For any complex number

w = c + id

, let

arg\left(w\right) \in ( - \pi, \pi ]

, where

i = \sqrt{- 1}

. Let

\alpha

and

\beta

be real numbers such that for all complex numbers

z = x + iy

satisfying

arg\left(\dfrac{z + \alpha}{z + \beta}\right) = \dfrac{\pi}{4}

, the ordered pair

( x, y )

lies on the circle

{x}^{2} + {y}^{2} + 5x - 3y + 4 = 0

Then which of the following statements is(are) TRUE? For any complex number w = c + id , let arg(w) ∈ ( - π, π ] , where i = - 1 . Let α and β be real numbers such that for

Options

A $\alpha = - 1$
B $\alpha\beta = 4$
C $\alpha\beta = - 4$
D $\beta = 4$

Correct answer

B. $\alpha\beta = 4$

Step-by-step solution

$arg\left(\dfrac{z + \alpha}{z + \beta}\right) = \dfrac{\pi}{4}$ represents an arc of a circle and \left(- \alpha, 0\right) \text{ and } \left(- \beta, 0\right) $subtends an angle$ \dfrac{\pi}{4} $on$ z $<img src="https://cdn-question-pool.Quantrex.app/pyq/jee_advanced/1555aaf8-429e-43fd-9cbf-40043700ee5c-d9e47279-9fdd-4864-a93f-6fcb96a260c0-image.png" style="width: 250px; height: 238px;" />Given,$ {x}^{2} + {y}^{2} + 5x - 3y + 4 = 0 $Centre \left(\dfrac{- 5}{2}, \dfrac{3}{2}\right), r = \sqrt{\dfrac{25}{4} + \dfrac{9}{4} - 4} = \dfrac{3}{\sqrt{2}}$

Angle made by $( - \alpha, 0 ) \text{\text{ and } } ( - \beta, 0 )$ at the centre $C$ is $\dfrac{\pi}{2}$

Using rotation of complex numbers, we get

$\dfrac{- \alpha - \left(- \dfrac{5}{2} + \dfrac{3}{2}i\right)}{- \beta - \left(- \dfrac{5}{2} + \dfrac{3}{2}i\right)} = 1 \cdot {e}^{i\dfrac{\pi}{2}} = i$

$- \alpha + \dfrac{5}{2} - \dfrac{3}{2}i = - i\beta + \dfrac{5}{2}i + \dfrac{3}{2}$

$\alpha - i\beta = 1 - 4i$

$∵ \alpha, \beta \in R$

$⇒ \alpha = 1, \beta = 4$

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