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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 $\left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, 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?
$(A)$ $\alpha=-1$ $(B)$ $\alpha \beta=4$ $(C)$ $\alpha \beta=-4$ $(D)$ $\beta=4$
$A,B$
$A,C$
$A,D$
$B,D$
Solution
$\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$ implies $z$ is
on arc and $(-\alpha, 0) \&(-\beta, 0)$ subtend $\frac{\pi}{4}$ on $z$.
And $z$ lies on $x^2+y^2+5 x-3 y+4=0$
So put $y=0$;
$x^2+5 x+4=0 \Rightarrow x=-1 ; x=-4$
Now, $\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4} \Rightarrow z+\alpha=(z+\beta)$. T. $e^{i \frac{\pi}{4}}$
So, $z+\beta=z+4 \Rightarrow \beta=4 \& z+\alpha=z+1 \Rightarrow \alpha=1$
