If $(3+4i)^{2025} = 5^{2023}(x+iy)$,then $\sqrt{x^2+y^2} = $

If $13 e^{i \tan ^{-1} \frac{5}{12}} = a + i b$,then the ordered pair $(a, b) =$

If $\frac{z - i}{z + i}$ $(z \ne -i)$ is a purely imaginary number,then $z \cdot \bar{z}$ is equal to

If $z = \frac{1}{2} - 2i$ is such that $|z+1| = \alpha z + \beta(1+i)$,where $i = \sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$,then $\alpha + \beta$ is equal to

If $x-iy = \sqrt{\frac{a-ib}{c-id}}$,prove that $(x^2+y^2)^2 = \frac{a^2+b^2}{c^2+d^2}$.

If $a > 0$ and $z = \frac{(1+i)^2}{a-i}$,where $i = \sqrt{-1}$,has a magnitude of $\frac{2}{\sqrt{5}}$,then $\bar{z}$ is