## [Coordinate method on the plane] $[$ Circles (other) $\quad]$

Given points $A(0 ; 0), B(-2 ; 1), C(3 ; 3), D(2 ;-1)$ and the circle $(x-1)^{2}+(y+3)^{2}=25$. Determine the location of these points: on the circle, inside the circle, or outside the circle.
Substituting the coordinates of the given points into the left side of the equation of the given circle, we will find the squares of the distances from these points to the center $Q(1; -3)$ of the circle:

$$
\begin{gathered}
Q A^{2}=(0-1)^{2}+(0+3)^{2}=1025, \\
Q D^{2}=(2-1)^{2}+(-1+3)^{2}=5<25
\end{gathered}
$$

Therefore, points $A$ and $D$ are located inside the circle, point $B$ is on the circle, and point $C$ is outside the circle.

## Answer

Points $A$ and $D$ are located inside the circle, point $B$ is on the circle, and point $C$ is outside the circle.

## Problem