For any complex number w = c + i d , let arg w ∈ ( - π , π ] , where i = - 1 . Let α…

$\arg \left(\frac{z+\alpha }{z+\beta }\right)=\frac{\pi }{4}$ represents an arc of a circle and $\left(-\alpha ,0\right) \& \left(-\beta ,0\right)$ subtends an angle $\frac{\pi }{4}$ on $z$

Given, $x^2+y^2+5x-3y+4=0$

Centre $\left(\frac{-5}{2},\frac{3}{2}\right), r=\sqrt{\frac{25}{4}+\frac{9}{4}-4}=\frac{3}{\sqrt{2}}$

Angle made by $(-\mathrm{\alpha },0) \& (-\mathrm{\beta },0)$ at the centre $C$ is $\frac{\pi }{2}$

Using rotation of complex numbers, we get

$\frac{-\alpha -\left(-\frac{5}{2}+\frac{3}{2}i\right)}{-\beta -\left(-\frac{5}{2}+\frac{3}{2}i\right)}=1·e^{i\frac{\pi }{2}}=i$

$-\alpha +\frac{5}{2}-\frac{3}{2}i=-i\beta +\frac{5}{2}i+\frac{3}{2}$

$\alpha -i\beta =1-4i$

$\because \alpha , \beta \in R$

$\Rightarrow \alpha =1, \beta =4$