[ Geometry of the complex plane ] [ Inscribed circle in an angle ]

Among the complex numbers $p$, satisfying the condition $|p-25 i| \leq 15$, find the number with the smallest argument.
All complex numbers satisfying the condition $[p-25 i \mid \leq 15$ are represented by points lying inside and on the circle with center at point $25 i$ and radius 15 (see figure). The smallest argument will be at the right point of tangency of the radius vector and the circle. Let's find the coordinates of the point $P$ of tangency $O P$ with the circle. From the right triangle $O P B, O P^{2}=O B^{2}-B P^{2}=25^{2}-15^{2}=20^{2}(O B=25, B P=15), P A \perp O B$. From the right triangle $O P B: \quad A P=B P \cdot O P / O B, \quad A O^{2}=O P^{2}-A P^{2}=256$.

![](https://cdn.mathpix.com/cropped/2024_05_06_3da1f4345ad15e82abd2g-22.jpg?height=709&width=666&top_left_y=0&top_left_x=707)

## Answer

$12+16 i$