If the points in the complex plane which satisfy the equations ${\log }_5\left(\right|\mathrm{z}|+3)-{\log }_{\sqrt{5}}\left|\right|\mathrm{z}|-1|=1$ and $\arg (\mathrm{z}-1-\mathrm{i})=\frac{\mathrm{\pi }}{4}$ is of the form (A₁ + iB₁), then the value of (A₁ + B₁) is equal to

${\log }_5\left(\frac{\left|\mathrm{z}\right|+3}{\left|\right|\mathrm{z}|-1{|}^2}\right)=1\Rightarrow \left|\mathrm{z}\right|+3=5\left|\right|\mathrm{z}|-1{|}^2------\left(\mathrm{i}\right)$

${\tan }^{-1}\left(\frac{\mathrm{y}-1}{\mathrm{x}-1}\right)=\frac{\mathrm{\pi }}{4}\dots \cdots \left(2\right)$

$\Rightarrow y-1=x-1\therefore y=x$

$\left|\mathrm{z}\right|+3=5\left|\right|\mathrm{z}|-1{|}^2\Rightarrow \sqrt{x^2+y^2}+3=5{\left|\sqrt{x^2+y^2}-1\right|}^2$

$$\begin{gathered} x\sqrt{2}+3=5\left(2x^2+1-2x\sqrt{2}\right) \\ \Rightarrow 10x^2-11x\sqrt{2}+2=0 \\ \Rightarrow 10x^2-10x\sqrt{2}-x\sqrt{2}+2=0 \\ \Rightarrow 10x\left(x-\sqrt{2}\right)-\sqrt{2}\left(x-\sqrt{2}\right)=0 \\ \Rightarrow \left(x-\sqrt{2}\right)\left(10x-\sqrt{2}\right)=0 \end{gathered}$$
$\mathrm{Only} (\sqrt{2},\sqrt{2}) \mathrm{satisfies}.$