For any complex number w=c+i d, 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+i y satisfying \arg (\frac{z+\alpha}{z+\beta})=\frac{\pi}{4}, the ordered pair (x, y) lies on the circle x^{2}+y^{2}+5 x-3 y+ 4=0. Then which of the following statements is (are) TRUE? [JEE Adv 2021 P1]

Complex Numbers

Section: Mathematics

Question : 10 of 36

Marks: +1, -0

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

)=‌

, the ordered pair (x,y) lies on the circle x2+y2+5x−3y+ 4=0.
Then which of the following statements is (are) TRUE?

[JEE Adv 2021 P1]

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