For any complex number \( w=c+i d \), let \( \arg (w) \in(-\pi, \pi] \), P where \( i=\sqrt{-1} ...

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For any complex number \( w=c+i d \), let \( \arg (w) \in(-\pi, \pi] \),
P where \( i=\sqrt{-1} \). Let \( \alpha \) and \( \beta \) be real numbers such that for

W) all complex numbers \( z=x+i y \) satisfying \( \arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4} \), 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?
(a) \( \alpha=-1 \)
(b) \( \alpha \beta=4 \)
(c) \( \alpha \beta=-4 \)
(d) \( \beta=4 \)
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