Problem 373

Let's find the coordinates of the points of intersection $A\left(x_{1} ; y_{1}\right)$ and $B\left(x_{2} ; y_{2}\right)$ of the given line and circle. For this, we will solve the system of equations

$$\left\{\begin{array}{l} y=3 x \\ (x+1)^{2}+(y-2)^{2}=25 \end{array}\right.$$

We obtain: $x_{1}=-1, y_{1}=-3, x_{2}=2, y_{2}=6$.

Using the formula for the distance between two points

$$A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}=\sqrt{(2-(-1))^{2}+(6-(-3))^{2}}=\sqrt{9+81}=3 \sqrt{10}$$

## Answer

$3 \sqrt{10}$.