If $\left|\mathrm{z}-25\mathrm{i}\right|\le 15$, then maximum arg (z)$-$ minimum arg $\overline{)\left(\mathrm{z}\right)}$ equals

Complex number z satisfy  $\left|\mathrm{z}-25\mathrm{i}\right|\le 15$

Therefore z lies  either on or inside the circle with centre (0.25) and radius=15

The complex numbers having least positive argument and maximum positive arguments in this region are the points of contact of tangents drawn from origin to the circle.

Here, $\theta =$ Least positive argument and - $\varphi =$ Maximum positive argument

In $∆\mathrm{OAC} \cos ({90}^{\circ }-\mathrm{\theta })=\frac{\mathrm{OA}}{\mathrm{OC}}=\frac{20}{25}$

$$\begin{gathered} \Rightarrow \mathrm{\alpha }={\cos }^{-1}\frac{4}{5} \\ \left|\mathrm{Max} \mathrm{Arg}\left(\mathrm{z}\right)-\mathrm{Min} \mathrm{Arg} \mathrm{z}\right|=\angle QOP=2 {\cos }^{-1}\frac{4}{5} \end{gathered}$$