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Complex Numbers question

2021 · Shift 1 · Q35

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PYQ Library
JEE Advanced
Mathematics
Complex Numbers
2021 · Shift 1 · Q35

Complex Numbers question

2021 · Shift 1 · Q35

JEE AdvancedMathematicsComplex NumbersMultiple correct+4 / −2

For any complex number w = c + id, let

\arg (w) \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( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 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?

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

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