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