Question

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 $\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, then 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. $\beta=4$

Answer 1

Answer: (B), (D)

$\alpha \beta=4$

Answer 2

Circle x2 + y2 + 5x – 3y + 4 = 0 cuts the real axis (x-axis) at (-4, 0), (-1, 0)

Clearly α = 1 and β = 4